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Course of Positive Philosophy - Tome 1
COURS
DE
PHILOSOPHIE POSITIVE.
ÉVERAT, IMPRIMEUR, RUE DU CADRAN, Nº 16.
[NOTE DU TRANSCRIPTEUR: Ce premier volume contient un grand nombre de
formules algébriques. Les correcteurs d'épreuve ont tenté de reproduire
ces formules tant bien que mal, cependant comme le format .txt ne se
prête pas très bien à cet exercice. Ces corrections pourront s'avérer
incompréhensibles pour la plupart des lecteurs, et possiblement
incorrectes pour les autres. Pour une version plus complète, et plus
specifies the reader will have great advantage to consult the HTML version of
this document.]
COURSE
OF
POSITIVE PHILOSOPHY,
By M. Auguste Comte,
FORMER STUDENT OF THE POLYTECHNIQUE SCHOOL.
TOME FIRST,
CONTAINING
GENERAL PRELIMINARIES AND
MATHEMATICAL PHILOSOPHY .
PARIS.
ROUEN FRÈRES, LIBRARIES-PUBLISHERS,
RUE DE L'ÉCOLE DE MÉDICINE, Nº 13.
BRUSSELS,
DEPOSIT OF THE FRENCH MEDICAL LIBRARY.
1830.
TO MY ILLUSTRATED FRIENDS
M. le Baron Fourier, Permanent Secretary
of the Royal Academy of
Sciences,
Professor GMD de
Blainville, Member of the
Royal Academy of Sciences,
In testimony of my respectful affection,
Auguste Comte,
Former student of the École Polytechnique.
AUTHOR WARNING.
This course, the general result of all my work since leaving
the École Polytechnique in 1816, was opened for the first time in April
1826. After a small number of sessions, a serious illness prevented me
from continuing at that time. an enterprise encouraged, from its birth,
by the votes of several learned men of the first order, among whom
I could henceforth quote MM. Alexandre de Humboldt, de Blainville and
Poinsot, members of the Académie des Sciences,
with sustained interest the exhibition of my ideas. I repeated this
entire course last winter, from January 4, 1829, in front of an
audience of which M. Fourier,
perpetual secretary of the Academy of Sciences, MM. de Blainville, Poinsot,
Navier, members of the same academy, MM. Professors Broussais,
Esquirol, Binet, etc., to whom I must publicly testify here my
gratitude for the way in which they received this new
philosophical attempt.
After having assured myself by such votes that this course could usefully
receive greater publicity, I felt it necessary, for this purpose,
to exhibit it this winter at the Athénée Royal in Paris,
open on December 9. The plan has remained completely the same.
Only the convenience of this establishment obliges me to restrict
the development of my course a little. They are fully reflected
in the publication that I am making today of my lessons, as
they took place last year.
To complete this historical notice, it is advisable to point out
, in relation to some of the fundamental ideas exposed
in this course, that I had presented them previously in the
first part of a work entitled _System of positive politics_,
printed in a hundred copies. in May 1822, and then reprinted in April
1824, to a larger number of copies. This first part
has not yet been formally published, but only communicated,
by way of printing, to a large number of European scholars and
philosophers. It will not be put into circulation definitively
until the second part which I hope to be able to publish at the end
of the year 1830.
I thought it necessary to note here the effective publicity of this
first work, because some ideas offering a certain analogy
with part of mine, are exposed, without any mention
of my research, in various works published subsequently, especially
with regard to the renovation of social theories.
different minds have been able, without any communication, as
the history of the human mind often shows , to arrive separately at
similar conceptions by dealing with the same class of work, I
nevertheless had to insist on the real anteriority of 'a work little
known to the public, so that one does not suppose that I have drawn the germ of
certain ideas from writings which are, on the contrary, more recent.
Several persons having already asked me for some clarifications
relative to the title of this course, I think it useful to indicate here, on this
subject, a summary explanation.
The expression _positive philosophy_ being constantly used
throughout the course of this course,
invariable, it seemed to me superfluous to define it other than by the
uniform use that I have always made of it. The first lesson, in particular,
can be regarded as a whole as the development of the
exact definition of what I call _ positive philosophy_. I regret,
however, that I have been obliged to adopt, in the absence of any other, a term
like that of _philosophie_, which has been so abused in a
multitude of different meanings. But the adjective _positive_ by which
I modify the meaning seems to me sufficient to remove, even at
first glance, any essential ambiguity, in those, at least, who
know its value well. I will therefore limit myself, in this warning,
to declare that I use the word _philosophy_ in the meaning given to it
by the ancients, and particularly Aristotle, as designating the
general system of human conceptions; and, by adding the word
_positive_, I announce that I consider this special way of
philosophizing which consists in considering the theories, in whatever order
of ideas, as having for object the coordination of the
observed facts , which constitutes the third and last state of
general philosophy, initially theological and then metaphysical,
as I explain from the first lesson.
There is, without doubt, a lot of analogy between my _positive philosophy_
and what English scholars understand, especially since Newton, by
_philosophie naturelle_. But I did not have to choose this last
name, nor that of _philosophie des sciences_ which
would perhaps be even more precise, because both do
not yet understand all the orders of phenomena. , while
_positive philosophy_, in which I understand the study of
social phenomena as well as of all others, denotes a
uniform way of reasoning applicable to all matters on which
the human mind can exercise. In addition, the expression _philosophie
naturelle_ is used, in England, to designate the set of
various observational sciences, considered even in their
most detailed specialties; instead that by _ positive philosophy_,
compared to _ positive sciences_, I mean only the proper study of the
generalities of the different sciences, conceived as subject to a
single method, and as forming the different parts of a
general plan of research. The term that I was led to construct is
therefore, at the same time, more extensive and more restricted than the denominations,
moreover analogous, as regards the fundamental character of the ideas, which one
could, at first glance, regard as equivalent.
Paris, December 18, 1829.
COURSE
OF
POSITIVE PHILOSOPHY.
FIRST LESSON.
SUMMARY. Statement of the purpose of this course, or general considerations on
the nature and importance of positive philosophy.
The object of this first lesson is to set out clearly the aim of the course,
that is to say to determine exactly the spirit in which will be
considered the various fundamental branches of
natural philosophy , indicated by the summary program that I introduced you.
Doubtless, the nature of this course can only be fully
appreciated, so as to be able to form a definitive opinion,
when the various parts have been successively developed.
This is the ordinary drawback of definitions relating to
very extensive systems of ideas, when they precede their exposition.
But the generalities can be conceived in two aspects, or as an
outline of a doctrine to be established, or as a summary of an established doctrine.
If it is only from this last point of view that they acquire
all their value, they nevertheless already have, under the first, an
extreme importance, in characterizing from the outset the subject to be
considered. The general circumscription of the field of our research,
traced with all the severity possible, is, for our mind, a
particularly indispensable preliminary in a study as vast
and so far as little determined as that of which we are going.
to occupy. It is in order to obey this logical necessity that I believe I
must indicate to you, from this moment, the series of
fundamental considerations which gave birth to this new course, and which will
moreover be specially developed, in the following, with all
the extension that the high importance of each of them demands.
In order to properly explain the true nature and proper character
of positive philosophy, it is essential to first take a
general look at the progressive course of the human mind,
considered as a whole: for no conception whatever is may be
well known only by its history.
By thus studying the total development of the
its various spheres of activity, from its simplest first development
to the present day, I believe I have discovered a great fundamental law,
to which it is subject by an invariable necessity, and which
seems to me to be able to be firmly established, either on the
rational evidence provided by the knowledge of our organization, or
on historical verifications resulting from a careful examination of the
past. This law consists in the fact that each of our
main conceptions , each branch of our knowledge, passes successively
through three different theoretical states: the theological or fictitious state;
the metaphysical, or abstract state; scientific state, or positive. In
In other words, the human mind, by its nature, successively employs
in each of its researches three methods of philosophizing, the
character of which is essentially different and even radically opposed:
first the theological method, then the metaphysical method, and
finally the positive method. Hence, three kinds of philosophies, or
general systems of conceptions on all phenomena, which
are mutually exclusive: the first is the necessary starting point of
human intelligence; the third, its fixed and final state: the
second is only intended to serve as a transition.
In the theological state, the human spirit essentially directing its
research towards the intimate nature of beings, the first and
final causes of all the effects which strike him, in a word, towards
absolute knowledge, the phenomena are represented as produced by
the direct and continuous action of more or less supernatural agents numerous,
whose arbitrary intervention explains all the apparent anomalies
of the universe.
In the metaphysical state, which is at bottom only a simple
general modification of the first, the supernatural agents are replaced by
abstract forces , true entities (personified abstractions) inherent
in the various beings of the world, and conceived as capable of generate by
themselves all the observed phenomena, including the
then to assign for each the corresponding entity.
Finally, in the positive state, the human mind recognizing
the impossibility of obtaining absolute notions, gives up seeking
the origin and destination of the universe, and knowing the
inner causes of phenomena, in order to only to discover, by
the well-combined use of reasoning and observation, their
effective laws , that is to say, their invariable relations of succession and
similitude. The explanation of facts, then reduced to its real terms,
is henceforth nothing more than the connection established between the various
particular phenomena and some general facts, including the progress of science.
tend more and more to decrease the number.
The theological system has reached the highest perfection of which it
is capable, when it substituted the providential action of a
single being for the varied play of the many independent divinities which had
been imagined initially. Likewise, the last term of the
metaphysical system consists in conceiving, instead of the various
particular entities , a single great general entity, _nature_, considered
as the single source of all phenomena. Likewise, the
perfection of the positive system, towards which it tends incessantly,
although it is very probable that it should never attain it, would be
to be able to represent all the various observable phenomena as
particular cases of a single general fact, such as that of
gravitation, for example.
This is not the place to demonstrate especially this
fundamental law of the development of the human mind, and to deduce from it
the most important consequences. We will deal with it directly, with
all the appropriate extension, in the part of this course relating to
the study of social phenomena [1]. I only consider it now
to determine with precision the true character of
positive philosophy , as opposed to the two other philosophies which have
successively dominated, until these last centuries, our whole system.
intellectual. As for the present, in order not to leave entirely without
demonstration a law of this importance, the applications of which will
present themselves frequently throughout the whole extent of this course, I must
confine myself to a rapid indication of the most sensitive general reasons
which can see the accuracy.
[Note 1: People who would immediately like
more
detailed explanations on this subject may usefully consult three articles in _Considerations
philosophiques sur les sciences et les savans_ that I
published in November 1825 in a collection entitled _the
Producteur_ (nos 7 , 8 and 10), and especially the first part
of my _System of positive politics_, addressed in April
1824 to the Academy of Sciences, and in which I recorded, for
the first time, the discovery of this law.]
In the first place, it suffices, it seems to me, to state such a law, so
that its correctness is immediately verified by all those who have
some deep knowledge of the general history of science. There
is not a single one, in fact, which has reached a
positive state today , that each one cannot easily imagine, in the past,
essentially composed of metaphysical abstractions, and, going
even further back , all- completely dominated by theological conceptions.
We will unfortunately have even more than one formal occasion to
recognize, in the various parts of this course, that the
most perfected sciences still preserve today some
very sensitive traces of these two primitive states.
This general revolution of the human mind can, moreover, be
easily ascertained today, in a very sensitive, although
indirect manner, by considering the development of
individual intelligence . The point of departure being necessarily the same in
the education of the individual as in that of the species, the various
principal phases of the first must represent the
fundamental epochs of the second. Now, each of us,
own history, does he not remember that he was successively, in terms
of his most important notions, _theologian_ in his childhood,
_metaphysician_ in his youth, and _physician_ in his virility? This
verification is easy today for all men at the level of
their century.
But, in addition to direct observation, general or individual, which proves
the correctness of this law, I must above all, in this
summary indication , mention the theoretical considerations which make it felt
necessary.
The most important of these considerations, drawn from the very nature of the
subject, consists in the need, at all times, for some theory
to link the facts together with the obvious impossibility for
the human mind at its origin to form theories from
observations.
All good minds have repeated, since Bacon, that there is no
real knowledge except that which is based on observed facts.
This fundamental maxim is evidently incontestable, if we
apply it, as it should, to the virile state of our intelligence.
But by referring to the formation of our knowledge, it is nonetheless
certain that the human mind, in its primitive state,
could not and should not have thought thus. Because, if on the one hand, any
positive theory must necessarily be founded on observations, it is
equally sensitive, on the other hand, that in order to engage in
observation our mind needs some theory. If in
contemplating the phenomena we did not immediately link them
to some principles, not only would it be impossible for us to
combine these isolated observations, and consequently to derive any
fruit from them, but we would even be entirely incapable of retaining them; and,
more often than not, the facts would remain unnoticed before our eyes.
Thus, pressed between the need to observe in order to form
real theories , and the no less imperative need to create
any theories for oneself in order to engage in constant observations, the human mind,
at birth, would find himself trapped in a vicious circle from which he
would never have had any way out, if he had not fortunately
opened up a natural outcome through the spontaneous development of
theological conceptions , which presented a rallying point to his efforts, and
fed his activity. Such is, independently of the high
social considerations which are attached to it and which I do not even have to
indicate at this moment, the fundamental motive which demonstrates the
logical necessity of the purely theological character of the primitive philosophy.
This necessity becomes even more sensitive by having regard to the
perfect suitability of theological philosophy with its own nature.
research on which the human mind in its childhood
so eminently concentrates all its activity. It is quite remarkable, in fact, that
the questions most radically inaccessible to our means, the
intimate nature of beings, the origin and the end of all phenomena,
are precisely those which our intelligence proposes itself above
all in this primitive state, all really soluble problems being
almost regarded as unworthy of serious meditation. We can
easily see the reason for this; for it is experience alone which has been able to furnish us with
the measure of our strength; and, if the man had not at first begun by
having an exaggerated opinion of them, they would never have been able to acquire all the
development of which they are susceptible. This is what our
organization demands . But, whatever the case, let us represent, as far as
possible, this disposition so universal and so pronounced, and
let us ask ourselves what reception would have received at such an epoch,
supposing it formed, the positive philosophy, of which the most high ambition
is to discover the laws of phenomena, and the first characteristic of which
is precisely to regard as necessarily forbidden to
human reason all those sublime mysteries, which
theological philosophy explains, on the contrary, with such admirable ease,
even in their slightest details.
It is the same when considering from the practical point of view the nature
research which primarily occupies the human mind. In this
respect, they offer man the so energetic attraction of an
unlimited empire to be exercised over the external world, seen as entirely
intended for our use, and as presenting in all its phenomena
intimate and continuous relations with our existence. . Now these
chimerical hopes , these exaggerated ideas of the importance of man in
the universe, which theological philosophy gives birth to, and which
the first influence of positive philosophy destroys without return, are, at
the origin, a indispensable stimulant, without which one could
certainly not conceive that the
Today we are so far removed from these
first dispositions , at least as regards most phenomena, that we have
difficulty in understanding exactly the power and the necessity of
similar considerations. Human reason is now mature enough
for us to undertake laborious scientific research,
without having in view any foreign goal capable of acting strongly on
the imagination, such as that proposed by astrologers or
alchemists. Our intellectual activity is sufficiently excited by
the pure hope of discovering the laws of phenomena, by the simple desire
to confirm or invalidate a theory. But it couldn't be
so in the infancy of the human mind. Without the attractive chimeras
of astrology, without the energetic disappointments of alchemy, for
example, where would we have drawn the constancy and the ardor necessary to
collect the long series of observations and experiences which have,
later , served as a basis for the first positive theories of both
classes of phenomena?
This condition of our intellectual development has long been keenly
felt by Kepler, for astronomy, and rightly
appreciated nowadays by Berthollet, for chemistry.
We can therefore see, by this set of considerations, that, if philosophy
positive is the true definitive state of human intelligence, the
one towards which it has always tended more and more, it nevertheless
necessarily had to use first, and for a long series of
centuries, either as a method, or as a provisional doctrine,
theological philosophy; a philosophy whose character is to be
spontaneous, and, by that very fact, the only one possible at the origin, the only one
also which could offer to our nascent mind a sufficient interest. It
is now very easy to feel that, in order to pass from this
provisional philosophy to the definitive philosophy, the human mind has
naturally had to adopt, as a transitory philosophy, the methods and
metaphysical doctrines. This last consideration is
essential to complete the general overview of the great law which I have
indicated.
It is easy to see, in fact, that our understanding, forced to
walk only by almost insensible degrees, could not pass
abruptly, and without intermediaries, from theological
philosophy to positive philosophy. Theology and physics are so profoundly
incompatible, their conceptions have a character so radically
opposed, that before renouncing the one to exclusively employ the
other, the human intelligence had to make use of
intermediate conceptions , of a bastard character, suitable, by that very fact, to operate
gradually the transition. Such is the natural destination of
metaphysical conceptions: they have no other real utility. By
substituting, in the study of phenomena, for the
directing supernatural action by a corresponding and inseparable entity, although the latter
was at first conceived only as an emanation of the former, man has
gradually become accustomed to to consider only the facts themselves, the notions
of these metaphysical agents having been gradually subtilized to the point
of being no longer, in the eyes of any upright mind, but the abstract names
of phenomena. It is impossible to imagine by what other process
our understanding could have passed frankly considerations
supernatural to purely natural considerations, from the
theological regime to the positive regime.
Having thus established, as far as I can do without entering into
a special discussion which would be out of place at this time, the
general law of the development of the human mind, as I understand it, it
will now be easy for us to determine with precision. the proper nature
of positive philosophy; which is the essential object of this speech.
We see from what precedes that the fundamental character of
positive philosophy is to regard all phenomena as subject
to invariable natural laws, the precise discovery of which and the
reduction to the least possible number is the goal of all our efforts,
considering as absolutely inaccessible and meaningless for us the
search for what are called the _causes_, either first or
final. It is needless to dwell much on a principle which has
now become so familiar to all who have made a somewhat
thorough study of the sciences of observation. Everyone knows, in fact, that,
in our positive explanations, even the most perfect, we do
not in any way pretend to expose the _causes_ generating the
phenomena, since we would never do anything but reduce the
difficulty, but only analyze exactly the circumstances
of their production, and to relate them to one another by
normal relations of succession and similitude.
Thus, to cite the most admirable example, we say that the
general phenomena of the universe are _explained_, as far as they
can be, by the law of Newtonian gravitation, because,
on the one hand, this beautiful theory shows us all the immense variety of
astronomical facts, as being but one and the same fact considered from
various points of view; the constant tendency of all molecules
towards each other in direct ratio of their masses, and in
inverse ratio of the squares of their distances; while, on the other hand, this
A general fact is presented to us as a simple extension of a phenomenon
which is eminently familiar to us, and which, by this alone, we regard
as perfectly known, the weight of bodies on the surface of the
earth. As for determining what this attraction and
gravity are in themselves , what are their causes, these are questions which
we all regard as insoluble, which are no longer the domain of
positive philosophy, and which we rightly abandon. to
the imagination of theologians, or to the subtleties of metaphysicians. The
clear proof of the impossibility of obtaining such solutions is
that, whenever we have tried to say something about it
truly rational, the greatest minds could only define these
two principles one by the other, by saying, for the attraction, that it
is nothing else than a universal gravity, and then, for the
gravity, it is simply in the earth's gravity. In
such explanations, which make smile when one claims to know the
intimate nature of things and the generation mode of phenomena are
, however, all that we can achieve more satisfactory, we
show as identical two orders of phenomena, which were so
long regarded as unrelated. No
righteous mind today seeks to go further.
It would be easy to multiply these examples, which will appear in crowds
throughout the course of this course, since such is now the spirit which
exclusively directs the great intellectual combinations. To
cite at this moment only one among contemporary works, I will choose
the fine series of researches of M. Fourier on the theory of
heat. It offers us the very sensitive verification of the
preceding general remarks . In fact, in this work, whose
philosophical character is so eminently positive, the most important
and precise laws of thermological phenomena are
revealed, without the author having inquired a single time about the nature
intimate with heat, without his having mentioned, other than to
indicate its emptiness, the controversy so agitated between the partisans of
calorific matter and those who make heat consist in the
vibrations of a universal ether. And nevertheless the highest questions,
many of which had never even been asked, are dealt with in this
work, palpable proof that the human mind, without throwing itself into
unapproachable problems, and by restricting itself to the search for an
order entirely positive, can find there inexhaustible nourishment for its
deepest activity.
After having characterized, as exactly as I am permitted to do
in this general outline, the spirit of positive philosophy, which this
whole course is intended to develop, I must now consider
what stage of her formation she has reached today, and what
remains to be done to complete her constitute.
To this end, we must first consider that the different branches of
our knowledge did not have to go through at equal speed the three
major phases of their development indicated above, nor,
consequently, to arrive simultaneously at the positive state. There exists, in this
respect, an invariable and necessary order, which our various kinds of
conceptions have followed and must have followed in their progress, and whose
exact consideration is the indispensable complement of the
fundamental law stated above. This order will be the special topic of the
next lesson. Let it suffice for us, for the present, to know that it
conforms to the diverse nature of phenomena, and that it is determined
by their degree of generality, simplicity and
reciprocal independence , three considerations which, although distinct, contribute to the
same end. Thus, astronomical phenomena first, as being the
most general, the simplest, and the most independent of all the
others, and successively, for the same reasons, the phenomena of
terrestrial physics properly so called, those of chemistry, and finally the
physiological phenomena, have been reduced to positive theories.
It is impossible to assign the precise origin of this revolution; for
it can be said with exactitude, as with all other great
human events, that it has been accomplished constantly and more and
more, particularly since the works of Aristotle and the school
of Alexandria, and then since the introduction of natural sciences
in Western Europe by the Arabs. However, since it is advisable
to fix a time to prevent the rambling of ideas, I will indicate
that of the great movement impressed on the human mind, two centuries ago,
by the combined action of the precepts of Bacon, of the conceptions of
Descartes, and the discoveries of Galileo, such as the moment when the spirit of
positive philosophy began to rule the world, in
obvious opposition to the theological and metaphysical spirit. It was
then, in fact, that the positive conceptions
clearly emerged from the superstitious and scholastic alliance which more
or less disguised the true character of all the previous works.
Since that momentous period, the movement of ascent of
positive philosophy, and the movement of decadence of
theological and metaphysical philosophy , have been extremely marked. They are
finally pronounced so much that it has become impossible today, to
all observers being aware of their century, of ignoring the
final destination of human intelligence for positive studies, as
well as its henceforth irrevocable estrangement from these vain
doctrines and for these provisional methods which could only suit
its first development. Thus, this fundamental revolution will
necessarily be accomplished in all its extent. If, therefore, it
still has some great conquest to make, some main branch of
the intellectual domain to invade, we can be sure that the
transformation will take place there, as it has taken place in all the
others. Because, it would obviously be contradictory to suppose that
the human mind, so disposed to the unity of method, retained
indefinitely, for a single class of phenomena, its primitive way
of philosophizing, when once it had succeeded in adopting for everything
else a new philosophical course, of an absolutely
opposite character .
Everything therefore boils down to a simple question of fact: does
positive philosophy , which over the last two centuries has gradually taken on such a
great extension, today embraces all orders of
phenomena? It is obvious that this is not, and that, consequently,
there still remains a great scientific operation to be carried out to give
to positive philosophy that character of universality, indispensable to
its final constitution.
In fact, in the four main categories of natural phenomena
enumerated earlier, astronomical, physical,
chemical and physiological phenomena , we notice an essential gap relating
to social phenomena, which, although understood implicitly among
physiological phenomena, deserve , either by their importance, or by
the difficulties specific to their study, to form a distinct category.
This last order of conceptions, which relates to the most
particular phenomena , the most complicated, and the most dependent on all the
others, must necessarily, by this alone, have been perfected more
slowly than all the preceding ones, even without having regard to the
more special obstacles which we shall consider later. However that may be, it
is evident that he has not yet entered the domain of
positive philosophy. The theological and metaphysical methods which, in
relation to all other kinds of phenomena, are
now no longer employed by anyone, either as a means of investigation,
or even only as a means of argument, are still, on the
contrary, exclusively used, under the one and the other relation, for
all that concerns social phenomena, although their insufficiency in
this respect is already fully felt by all good minds, weary
of these vain endless disputes between divine right and the
sovereignty of the people.
Here then is the great, but obviously the only gap which it is a question of
filling in order to complete the constitution of positive philosophy. Now
that the human mind has founded celestial physics, terrestrial physics,
either mechanical or chemical; organic physics, whether plant
or animal, it remains for him to complete the system of
observational sciences by founding _social physics_. Such is today,
in several capital respects, the greatest and most pressing
need of our intelligence: such is, I dare say it, the first aim of
this course, its special aim.
The conceptions which I will attempt to present in relation to the study of
social phenomena, and of which I hope that this discourse already lets one
glimpse the germ, cannot have as their object to
immediately give to social physics the same degree of perfection as to
earlier branches of natural philosophy, which would
obviously be chimerical, since these already offer
an extreme inequality in this respect, which is inevitable. But they will be
intended to imprint on this last class of our knowledge, that
positive character already assumed by all the others. If this condition
is once really fulfilled, the modern philosophical system
will finally be founded as a whole; for no observable phenomenon
can obviously fail to fall into some one of the five major
categories from then on established of astronomical, physical,
chemical, physiological and social phenomena . All our
fundamental conceptions having become homogeneous, philosophy will be
definitively constituted in a positive state; without ever being able to change
its character, it will only have to develop itself indefinitely by the
ever increasing acquisitions which will inevitably result from
new observations or deeper meditations. Having thereby acquired
the character of universality which it still lacks, philosophy
positive will become capable of replacing itself entirely, with all its
natural superiority, for theological philosophy and
metaphysical philosophy , of which this universality is today the only
real property , and which, deprived of such a motive of preference, does not will have more for
our successors than a historical existence.
The special purpose of this course being thus stated, it is easy to understand
its second purpose, its general purpose, which makes it a course in
positive philosophy , and not just a course in social physics.
Indeed, the foundation of social physics finally completing the
system of natural sciences, it becomes possible and even necessary
to summarize the various knowledge acquired, then reached a
fixed and homogeneous state, in order to coordinate them by presenting them as
so many branches of a single trunk, instead of continuing to
conceive them only as so many isolated bodies. It is for this purpose that
before proceeding to the study of social phenomena I will consider
successively, in the encyclopedic order announced above, the
various positive sciences already formed.
It is superfluous, I think, to warn that there could be no question here
of a series of special courses on each of the principal branches of
natural philosophy. Not to mention the material duration of a
a similar undertaking, it is clear that such a claim would be
untenable on my part, and I believe I can add on the part of
anyone, in the present state of human education. On the
contrary, a course of the nature of the latter requires, in order to be
properly understood, a preliminary series of special studies on the
various sciences which will be considered there. Without this condition, it is very
difficult to feel and impossible to judge the
philosophical reflections of which these sciences will be the subjects. In short, it is a
_Course in positive philosophy_, and not in positive sciences, that I
propose to do. It is only a question here of considering each
fundamental science in its relation to the all
entier, et quant à l'esprit qui la caractérise, c'est-à-dire, sous le
double rapport de ses méthodes essentielles et de ses résultats
major positive system . Most often even I will have to confine myself to mentioning these
last ones according to the special knowledge to try to appreciate
their importance.
In order to summarize the ideas relative to the dual purpose of this course, I
must observe that the two objects, one special, the other general,
which I propose to myself, although distinct in themselves, are necessarily
inseparable. For, on the one hand, it would be impossible to conceive of a course
in positive philosophy without the foundation of social physics,
since it would then lack an essential element, and that, by that
alone,
must be its main attribute, and which distinguishes our
present study from the series of special studies. On the other hand, how can we
proceed with safety in the positive study of social phenomena, if
the mind is not first prepared by the careful consideration of the
positive methods already judged for the less complicated phenomena, and
equipped, moreover, , of knowledge of the main laws of
previous phenomena , which all influence, in a more or less direct way,
on social facts?
While not all the basic sciences inspire the
common mind with equal interest, there is none that should be overlooked
in a study like the one we are undertaking.
importance for the happiness of the human species, all are certainly
equivalent, when considered in depth. Those,
moreover, whose results present, at first sight, of less
practical interest, are eminently recommended, either by the greatest
perfection of their methods, or as being the
indispensable foundation of all the others. This is a consideration to which
I will have special occasion to return to in the next lesson.
To prevent, as much as possible, all the false interpretations
that it is legitimate to fear about the nature of a course as new
as this one, I must add summarily to the preceding explanations
some considerations directly relating to this universality of
special knowledge, which thoughtless judges might regard
as the tendency of this course, and which is so rightly considered
as quite contrary to the true spirit of
positive philosophy . These considerations will have, moreover, the more
important advantage of presenting this spirit under a new point of view, apt to
complete the elucidation of its general notion.
In the primitive state of our knowledge there is no
regular division among our intellectual labors; all the sciences are
cultivated simultaneously by the same minds. This mode of organization
human studies, at first inevitable and even indispensable, as
we shall see later, change little by little
as the various orders of concept develop. By a law, the
necessity of which is evident, each branch of the scientific system
insensibly separates from the trunk, when it has grown sufficiently to
include an isolated culture, that is to say when it has reached this
point of to be able to occupy by itself the permanent activity of a few
intelligences. It is to this distribution of the various kinds of
research between different orders of scientists that we obviously owe
the remarkable development which has finally taken place in our days.
a class distinct from human knowledge, and which makes manifest
the impossibility, among the moderns, of that universality of
special research , so easy and so common in ancient times. In short,
the increasingly perfected division of intellectual labor is
one of the most important characteristic attributes of
positive philosophy .
But, while recognizing the prodigious results of this division,
while seeing in it henceforth the true fundamental basis of
the general organization of the learned world, it is impossible, on the other
hand, not to be struck by the capital inconveniences. that it generates,
in its present state, by the excessive particularity of the ideas which
exclusively occupy each individual intelligence. This unfortunate
effect is undoubtedly inevitable up to a certain point, as it is inherent
in the very principle of division; that is to say that, by any measure
whatsoever, we will never succeed in equaling the
ancients in this respect , among whom such superiority was due above all only to the little
development of their knowledge. We can nevertheless, it
seems to me , by suitable means, avoid the most pernicious effects of
exaggerated specialty, without harming the invigorating influence of the
separation of research. It is urgent to deal with it seriously;
because these inconveniences, which, by their nature, tend to increase without
cease, begin to become very sensitive. By everyone's admission, the
divisions, established for the greatest perfection of our work, between
the various branches of natural philosophy, are ultimately
artificial. Let us not forget that, notwithstanding this admission, it is already very
small in the learned world the number of intelligences embracing in
their conceptions the very whole of a single science, which is
however in turn only a part of it. 'a big whole. Most of them
already limit themselves entirely to the isolated consideration of a more or
less extensive section of a given science, without being much concerned with the
relation of these particular works with the general system of
positive knowledge. Let us hasten to remedy the evil, before it
becomes more serious. Let us fear that the human mind ends up getting
lost in the work of detail. Let us not hide from ourselves that this is
essentially the weak side through which the partisans of
theological philosophy and of metaphysical philosophy can still
attack positive philosophy with some hope of success.
The real means of stopping the deleterious influence, the
intellectual future of which seems threatened, owing to too great a specialization
of individual research, could obviously not be to return to
this ancient confusion of work, which would tend to make people downgrade
the human spirit, and which, moreover, has now fortunately
become impossible. On the contrary, it consists in perfecting
the division of labor itself. It suffices, in fact, to make
the study of scientific generalities one more major specialty.
Let a new class of savants, prepared by a suitable education,
without devoting themselves to the special cultivation of any particular branch of
natural philosophy, only occupy themselves, considering the various
positive sciences in their present state, in determining exactly
the mind of each of them, to discover their relationships and their
sequence, to summarize, if possible, all their own principles
in a smaller number of common principles, by constantly conforming
to the fundamental maxims of the positive method. That at the same time, the
other savants, before devoting themselves to their respective specialties,
be made able henceforth, by an education relating to the whole
of the positive knowledge, to immediately benefit from the enlightenment
spread by these savans dedicated to the study generalities, and
reciprocally to correct their results, a state of affairs to which
current scholars are visibly approaching day by day. These two
great conditions once fulfilled, and it is obvious that they can
be, the division of labor in the sciences will be pushed forward, without
no danger, as far as the development of the various orders of
knowledge will require. A distinct class, incessantly controlled
by all the others, having the proper and permanent function of linking
each new particular discovery to the general system, we no
longer have to fear that too much attention to detail will
never prevent us from perceiving the together. In short, the modern organization of
the learned world will therefore be completely founded, and will only have to
develop indefinitely, always retaining the same character.
To thus form from the study of scientific generalities a section
distinct from great intellectual work, it is simply to extend
the application of the same principle of division which has successively separated
the various specialties; for, as long as the different
positive sciences were little developed, their mutual relations could
not be of sufficient importance to give rise, at least
permanently, to a particular class of work, and at the same
time the necessity of this new study was much less urgent.
But today each of the sciences has taken enough
extension separately so that the examination of their mutual relations can give
rise to continued work, at the same time that this new order of study
becomes essential to prevent the dispersion of
human conceptions. .
Such is the way in which I conceive the destination of
positive philosophy in the general system of the positive sciences properly
so called. At least that is the aim of this course.
Now that I have tried to determine, as exactly as it was
possible for me to do, in this first overview, the general spirit of a
course in positive philosophy, I believe I must, in order to print
everything its character, to point out quickly the principal
general advantages which such work can have, if the essential conditions
are suitably fulfilled, relative to the progress of the
human mind . I will reduce this last order of considerations to the indication of
four fundamental properties.
First, the study of positive philosophy, by considering the
results of the activity of our intellectual faculties, provides us with the
only true rational means of bringing to light the logical laws of
the human mind, which have hitherto been sought by ways so little
adapted to reveal them.
To properly explain my thinking in this regard, I must first
recall a philosophical conception of the greatest importance,
expounded by M. de Blainville in the beautiful introduction to his
_Principes generales d'une anatomie comparée_. It consists in the fact that every
active being, and especially every living being, can be studied, in
all its phenomena under two fundamental relations, under the relation
static and dynamically, that is to say as capable of acting and
as acting effectively. It is clear, in fact, that all the
considerations which can be presented will necessarily fall into
one or the other mode. Let us apply this luminous fundamental maxim to
the study of intellectual functions.
If we consider these functions from the static point of view, their study
can consist only in the determination of the organic conditions on
which they depend; it thus forms an essential part of
anatomy and physiology. Considering them from the
dynamic point of view , everything boils down to studying the effective course of the mind.
human in practice, by examining the processes actually employed to
obtain the various exact knowledge which he has already acquired, which
essentially constitutes the general object of positive philosophy,
as I have defined it in this speech. In short, considering all
scientific theories as so many great logical facts, it is
only by the careful observation of these facts that one can
rise to the knowledge of logical laws.
These are obviously the only two general ways, complementary
to each other, by which we can arrive at some
true rational notions on intellectual phenomena. We see that,
in no respect is there room for this illusory psychology, the
last transformation of theology, which we are trying so vainly to
revive today, and which, without worrying about the
physiological study of our intellectual organs , nor from the observation of the
rational processes which effectively direct our various
scientific researches , claims to arrive at the discovery of the fundamental laws of
the human mind, by contemplating it in itself, that is to say by
completely disregarding and causes and effects.
The preponderance of positive philosophy has successively become
such since Bacon; she took today, indirectly, such a large
ascending over the very minds which have remained the most foreign to
its immense development, that the metaphysicians given over to the study of
our intelligence could only hope to slow the decadence of their
so-called science by changing their minds to present their doctrines as
being also based on observation of the facts. To this end, they have
imagined, in recent times, to distinguish, by a very
singular subtlety , two kinds of observations of equal importance, one
external, the other internal, and the last of which is only
intended for the study of intellectual phenomena. This is not the
place to enter into the special discussion of this fundamental fallacy. I
I must confine myself to indicating the main consideration which
clearly proves that this so-called direct contemplation of the mind by
itself is a pure illusion.
It was believed, a short time ago, to have explained vision by
saying that the luminous action of bodies determines on the retina
representative pictures of exterior shapes and colors. To this
physiologists have rightly objected that if it was like
_images_ that light impressions acted, it would take another
eye to look at them. Is it not even more strongly the same
in the present case?
It is evident, in fact, that, by an invincible necessity, the mind
human can directly observe all phenomena, except his
own. Because, by whom would the observation be made? It is conceivable,
relative to moral phenomena, that man can observe
himself in relation to the passions which animate him, by this
anatomical reason , that the organs which are their seat are distinct from those
intended for the observing functions. . Even though everyone has had
occasion to make such remarks about him, they can
obviously never have a great scientific importance, and the
best means of knowing the passions will always be to
observe them outside; because every very pronounced state of passion, that is to say
precisely the one which it would be most essential to examine is
necessarily incompatible with the state of observation. But, as to
observe the intellectual phenomena in the same way while they are being
performed, there is manifest impossibility. The thinking individual can
not be divided into two, one of whom would reason, while the other would
watch him reason. The observed organ and the observing organ being,
in this case, identical, how could the observation take place?
This so-called psychological method is therefore radically null in
principle. Also, consider what deeply
contradictory processes it immediately leads to! On the one hand, we have you
recommends that you isolate yourself, as much as possible, from any
external sensation , above all, you must refrain from any intellectual work;
for if you were only busy doing the simplest calculation,
what would become of the _internal_ observation? On the other hand, after
having, at last, by dint of precautions, reached this perfect state of
intellectual sleep, you will have to occupy yourself with contemplating the
operations which will be carried out in your mind, when
nothing more happens there. ! Our descendants will undoubtedly see such
pretensions transported one day on the scene.
The results of such a strange way of proceeding are perfectly
in accordance with the principle. In the two thousand years that metaphysicians have
cultivated psychology in this way, they have not yet been able to agree on a single
intelligible and firmly established proposition. They are, even
today, divided into a multitude of schools which constantly dispute
over the first elements of their doctrines. The_inner observation_
generates almost as many divergent opinions as there are individuals
believing to be engaged in it.
The true scientists, the men dedicated to positive studies, are
still vainly asking these psychologists to cite a single
real discovery, large or small, which is due to this method so
vaunted. This does
absolutely without any result relative to the general progress of our
knowledge, independently of the eminent service which they rendered in
supporting the activity of our intelligence, at the time when it
could not have more substantial nourishment. But we can affirm that
everything which, in their writings, does not consist, according to the judicious
expression of an illustrious positive philosopher (M. Cuvier), in metaphors
taken for reasonings, and presents some true notion,
instead of coming from of their pretended method, has been obtained by
effective observations on the progress of the human mind, to which
the development of the sciences must have given rise from time to time.
Even still, these notions so clearly-sown, proclaimed with so
much emphasis, and which are due only to the infidelity of psychologists to their
pretended method, are they most often found either very exaggerated,
or very incomplete. , and much inferior to the remarks already made without
ostentation by the scholars on the procedures they employ. It would be
easy to cite striking examples of this if I did not fear to grant
too much extension here to such a discussion: see, among other things, what
happened with the theory of signs.
The considerations which I have just indicated, relative to
logical science , are still more manifest, when they are transferred to
logical art .
Indeed, when it is a question, not only of knowing what the
positive method is, but of having a sufficiently clear and
deep enough knowledge of it to be able to make effective use of it, it is in action. that it
must be considered; it is the various great applications already
verified that the human mind has made of it that should be studied. In short
, it is obviously only by the philosophical examination of the sciences
that it is possible to achieve this. The method is not likely to be
studied separately from the research in which it is employed; or, at least, it
is only a dead study, incapable of fertilizing the spirit which
indulges in it. All that
abstractly, it is reduced to generalities so vague that they can
not have any influence on the intellectual regime. When we have
well established, as a logical thesis, that all our knowledge must be
founded on observation, that we must proceed sometimes from facts to
principles, and sometimes from principles to facts, and some other
similar aphorisms, we know much less clearly the method
than someone who has studied, in a somewhat thorough way, a single
positive science, even without philosophical intention. It is for having
ignored this essential fact that our psychologists are led to take
their reveries for science, believing they understand the method.
positive for having read Bacon's precepts or
Descartes' speech .
I do not know if, later, it will become possible to give _a priori_ a
real method course completely independent of the
philosophical study of the sciences; but I am quite convinced that this cannot be
executed today, the great logical processes not yet
being able to be explained with sufficient precision separately from their
applications. I dare to add, moreover, that even if such an
enterprise could be carried out in the sequel, which, in fact, can be
conceived, it would never be, however, except by the study of the
regular applications of scientific processes that 'we could
to succeed in forming a good system of intellectual habits; which
is nevertheless the essential aim of the study of the method. I do not
need to dwell further at this time on a subject which will come up
frequently throughout this course, and on which I
will especially present new considerations in the next
lesson.
Such must be the first great direct result of
positive philosophy , the manifestation by experience of the laws which
our intellectual functions follow in their accomplishment, and, consequently, the
precise knowledge of the general rules suitable for proceeding
surely in the search for truth.
A second consequence, no less important, and of a much more
pressing interest , which
the establishment of the positive philosophy defined in this discourse necessarily intended to produce today ,
is to preside over the general overhaul of our education system.
Indeed, good minds unanimously recognize the need
to replace our European education, still essentially
theological, metaphysical and literary, by a _positive_ education, in
accordance with the spirit of our time, and adapted to the needs of
modern civilization. The various attempts which have multiplied
more and more over the past century, particularly in recent times,
to spread and constantly increase positive instruction, and
which the various European governments have always
eagerly associated themselves with when they have not taken the initiative,
sufficiently testify that, on all sides, the spontaneous feeling of this
need. But, while assisting as much as possible these useful
enterprises, we must not conceal the fact that, in the present state of
our ideas, they are in no way capable of attaining their
principal aim , the fundamental regeneration of general education. Because, the
exclusive specialty, the too pronounced isolation which still characterizes
our way of conceiving and cultivating the sciences, influences
necessarily to a high degree on how to display them in
teaching. If a good mind now wishes to study the
principal branches of natural philosophy, in order to form a
general system of positive ideas, he will be obliged to study
each of them separately in the same fashion and in the same way. same detail as if he
wanted to become specially or astronomer, or chemist, etc .; which
makes such an education almost impossible and necessarily very
imperfect, even for the highest intelligences placed in
the most favorable circumstances. Such a way of proceeding would
therefore be altogether chimerical in relation to general education. And
nevertheless this one absolutely requires a set of positive conceptions
on all the great classes of natural phenomena. It is such an
ensemble which must henceforth become, on a more or less
extensive scale, even among the popular masses, the permanent basis of all
human combinations; which must, in a word, constitute the
general spirit of our descendants. For natural philosophy to be able to
complete the regeneration, already so prepared, of our
intellectual system , it is therefore essential that the different sciences of
which it is composed, presented to all intelligences as the
various branches of a single trunk, be reduced by 'first to what
constitutes their spirit, that is to say, their principal methods and
their most important results. Only in this way
can science education among us become the basis for a new
and truly rational general education . That following this
fundamental instruction are added the various special scientific studies,
corresponding to the various special educations which are to succeed
general education, this obviously cannot be doubted. But the
essential consideration that I wanted to indicate here consists in the fact that
all these specialties, even painfully accumulated, would
necessarily be insufficient to really renew the system of
our education, if they did not rest on the preliminary basis of this
general education, the direct result of the positive philosophy defined
in this discourse.
Not only is the special study of scientific generalities
intended to reorganize education, but it must also contribute to the
particular progress of the various positive sciences; which constitutes
the third fundamental property that I have proposed to point out.
Indeed, the divisions that we establish between our sciences, without
being arbitrary, as some believe, are essentially
artificial. In reality, the subject of all our research is one;
we only share it with a view to separating the difficulties in order
to better resolve them. It follows more than once that, contrary to
our classical distributions, important questions would require a
certain combination of several special points of view, which can
hardly take place in the present constitution of the learned world; which
leaves the possibility of leaving these problems unresolved much longer
than necessary. Such a disadvantage must present itself
especially for the most essential doctrines of each
positive science in particular. We can easily cite
very striking examples , which I will carefully point out as the
natural development of this course will introduce them to us.
I could cite an eminently memorable example of this in the past,
considering Descartes' admirable conception of
analytical geometry . This fundamental discovery, which changed the face of
mathematical science, and in which we must see the real germ
of all subsequent great progress, what is it other than the
result of a rapprochement established between two sciences, conceived
until then in an isolated way? But the observation will be more decisive
by focusing it on questions still pending.
I will limit myself here to choosing in chemistry, the so important doctrine
defined proportions. Certainly, the memorable discussion raised
in our day, relative to the fundamental principle of this theory, can
not yet, whatever appearances, be regarded as
irrevocably ended. Because, it is not there, it seems to me, a simple
question of chemistry. I believe I can advance that, in order to obtain
a truly definitive decision in this regard, that is to say, to determine whether
we must regard as a law of nature that molecules
necessarily combine in fixed numbers, it will be essential to
combine the chemical point of view with the physiological point of view. What
indicates it is that, by the admission even of the illustrious chemists who have
the most powerfully contributed to the formation of this doctrine, one can
say at the most that it is constantly verified in the composition of
inorganic bodies; but it is at least as constantly
lacking in organic compounds, to which it seems up to now
quite impossible to extend it. However, before setting up this theory into
a really fundamental principle, should we not first be
aware of this immense exception? Would it not be due to this same
general character, specific to all organized bodies, which means that,
in any of their phenomena, there is no reason to conceive of
invariable numbers ? Anyway, a whole new order of
considerations, belonging equally to chemistry and physiology,
is evidently necessary to finally decide, in some way
, this great question of natural philosophy.
I believe it appropriate to indicate here again a second example of the same
nature, but which, relating to a much more
particular subject of research , is still more conclusive in showing the
special importance of positive philosophy in the solution of the questions which
require the combination of several sciences. I also take it in
chemistry. This is the still undecided question, which consists of
determining whether nitrogen should be looked at, in the present state of our
knowledge, as a simple body or as a compound body. You
know by what purely chemical considerations the illustrious Berzelius
succeeded in balancing the opinion of almost all current chemists,
relative to the simplicity of this gas. But what I must not
neglect to point out particularly is the influence exerted
on this subject on the mind of M. Berzélius, as he himself makes the
precious admission, by this physiological observation, that animals that
feed on non-nitrogenous matter contain in the composition of
their tissues just as much nitrogen as carnivorous animals. It is
clear, indeed, from this, that in order really to decide whether nitrogen
is or is not a simple body, it will necessarily be necessary to bring into play
physiology, and to combine with chemical considerations properly
so called, a series of new researches on the relation between the
composition of living bodies and their mode of nourishment.
It would now be superfluous to multiply further the examples of
these problems of multiple nature, which can only be solved by
the intimate combination of several sciences cultivated today in a
completely independent way. Those which I have just quoted suffice
to make one feel, in general, the importance of the function which must
fulfill in the improvement of each natural science in
especially positive philosophy, immediately destined to organize
in a permanent way such combinations, which could not be
formed properly without it.
Finally, a fourth and last fundamental property which I must
point out from this moment in what I have called
positive philosophy , and which must undoubtedly merit
general attention to it more than any other , since it is today most important
for practice is that it can be considered as the only
solid basis for the social reorganization which must end the state of crisis
in which the most important nations have so long found themselves.
civilized. The last part of this course will be specially devoted to
establishing this proposition, developing it to its full extent.
But the general outline of the large picture which I have undertaken to indicate
in this speech would lack one of its most
characteristic elements , if I neglected to point out here such an
essential consideration .
A few very simple reflections will suffice to justify what
such a qualification initially appears to be too ambitious.
It is not to the readers of this work that I shall ever believe that I must
prove that ideas govern and upset the world, or, in
other words, that the whole social mechanism ultimately rests on
opinions. They know above all that the great political and moral crisis of
present-day societies is due, in the last analysis, to
intellectual anarchy . Our gravest evil consists, in fact, in this
profound divergence which now exists between all minds in
relation to all the fundamental maxims of which fixity is the
first condition of a true social order. As long as
individual intelligences have not adhered by
unanimous consent to a certain number of general ideas capable of forming a
common social doctrine, it cannot be concealed that the state of
nations will necessarily remain essentially revolutionary.
despite all the political palliatives that may be adopted, and will
really only include provisional institutions. It is also
certain that if this reunion of minds in the same communion of
principles can once be obtained, the suitable institutions
will necessarily flow from it, without giving rise to any serious shock, the
greatest disorder already being dispelled by this single fact. It is therefore here
that the attention of all those who feel
the importance of a truly normal state of affairs must be paid mainly .
Now, from the high point of view into which the
various considerations indicated in this discourse have gradually placed us , it is easy to
times and to characterize clearly in its intimate depth the
present state of societies, and to deduce from it by what way it can be changed
essentially. By relating myself to the fundamental law enunciated at the
beginning of this speech, I believe I can summarize exactly all
the observations relating to the present situation of society, by
simply saying that the present disorder of intelligences depends, in the
last analysis, on employment. simultaneous of three
radically incompatible philosophies : theological philosophy,
metaphysical philosophy and positive philosophy. It is clear, in fact, that if
any one of these three philosophies actually obtained a
universal and complete preponderance, there would be a
determined social order , while the evil consists above all in the absence of any
real organization. It is the coexistence of these three
opposing philosophies that absolutely prevents agreement on any essential point.
Now, if this view is correct, it is only a question of knowing
which of the three philosophies can and must prevail by the nature of
things; every sane man must then, whatever may have been
his particular opinions before the analysis of the question, strive to
contribute to his triumph. The research being once reduced to these
simple terms, it does not seem likely to remain uncertain for long;
for it is evident, for all sorts of reasons,
some of the main ones of which I have indicated in this speech, that positive philosophy
alone is destined to prevail according to the ordinary course of things.
It alone has been, for a long succession of centuries, constantly in
progress, while its antagonists have been constantly in decline.
Whether rightly or wrongly, it doesn't matter; the general fact is
incontestable, and it suffices. We can deplore it, but not destroy it,
nor consequently neglect it, under penalty of indulging in only
illusory speculations. This general revolution of the human mind
is today almost entirely accomplished: it does not remain, as
I have explained that to complete positive philosophy by including therein
the study of social phenomena, and then to summarize it in a single body
of homogeneous doctrine. When this double work is sufficiently advanced,
the definitive triumph of positive philosophy will take place spontaneously,
and will restore order in society. The preference so pronounced that
almost all minds, from the most elevated to the most
vulgar, today give to positive knowledge of
vague and mystical conceptions, is enough foreshadowing the reception which
this philosophy will receive , when it has acquired the the only quality which it
still lacks, a character of suitable generality.
To sum up, theological philosophy and metaphysical philosophy are
today competing for the task, too superior to the forces of both
, of reorganizing society: it is between them alone that
the struggle still exists, under this report. Positive philosophy has
hitherto intervened in the dispute only to criticize them
both, and it has done so well enough to discredit them
entirely. Let us finally put it in a position to take an active role, without
worrying about any longer debates that have become unnecessary. Completing the
vast intellectual operation begun by Bacon, Descartes and
Galileo, let us directly construct the system of general ideas that this
philosophy is henceforth destined to prevail indefinitely in
the human species, and the revolutionary crisis which torments
civilized peoples will be essentially over.
These are the four main points of view from which I have thought it
necessary to indicate from this moment the salutary influence of
positive philosophy , to serve as an essential complement to the general definition
which I have tried to set out.
Before concluding, I wish to draw attention for a moment to a
final reflection which seems to me appropriate to avoid, as far as
possible, that one forms in advance an erroneous opinion of the nature of this
course.
In assigning the goal of positive philosophy to summarize in a single
body of homogeneous doctrine all the knowledge acquired
relative to the different orders of natural phenomena, it was far
from my thought to want to proceed with the general study of these phenomena in
considering them all as various effects of a single principle, as
subject to one and the same law. Although I must treat
this question specially in the next lesson, I think it
right now to declare it, in order to prevent the
very ill-founded reproaches which might be addressed to me by those who, on a false overview,
classify this course. among these attempts at a universal explanation
which we see daily hatch from minds entirely
foreign to scientific methods and knowledge. This is not
about anything like that; and the development of this course will furnish the
manifest proof of it to all those in whom the clarifications contained
in this discourse might have left some doubts in this respect.
In my deep personal conviction, I consider these enterprises
of universal explanation of all phenomena by a single law
as eminently chimerical, even when they are attempted by
the most competent intelligences. I believe that the means of the
human mind are too weak, and the universe too complicated for such
scientific perfection is never within our reach, and I think,
moreover, that we generally form a very exaggerated idea of the
advantages which would necessarily result from it, if it were possible.
In any case, it seems obvious to me that, given the present state of our
knowledge, we are still far too far away for
such attempts to be reasonable before a
considerable lapse of time . Because, if we could hope to achieve this, it could only
be, in my opinion, by linking all natural phenomena to
the most general positive law that we know, the law of
gravitation, which already links all phenomena. astronomical at a party
those of terrestrial physics. Laplace has effectively exposed a
conception by which we could see in
chemical phenomena only simple molecular effects of
Newtonian attraction , modified by the shape and mutual position of atoms.
But, apart from the indeterminacy in which
this conception would probably always remain , by the absence of essential data
relating to the intimate constitution of bodies, it is almost certain that
the difficulty of applying it would be such that we would be obliged to
maintain , as artificial, the division now established as
natural between astronomy and chemistry. So Laplace did not
presented this idea only as a mere philosophical game, incapable
of actually exerting any useful influence on the progress of
chemical science. There is more, by the way; for, even supposing that
this insurmountable difficulty has been overcome, we would not yet have reached
scientific unity, since it would then be necessary to attempt to relate
all physiological phenomena to the same law; which, admittedly,
would not be the least difficult part of the business. And, nevertheless,
the hypothesis which we have just traversed would be, all things considered, the
most favorable to this much desired unity.
I don't need greater details to convince myself that
the aim of this course is by no means to present all
natural phenomena as being basically identical, except for the variety of
circumstances. Positive philosophy would doubtless be more perfect
if it could be so. But this condition is by no means
necessary for its systematic formation, nor for the realization of the
great and happy consequences which we have seen it destined to
produce. There is no unity indispensable for this except the unity of
method, which can and must obviously exist, and is already
established for the most part. As for the doctrine, it need not
be one; it suffices that it be homogeneous. It is therefore under the
double point of view of the unity of the methods and the homogeneity of the
doctrines that we will consider, in this course, the various classes
of positive theories. While tending to reduce as much as possible the
number of general laws necessary for the positive explanation of
natural phenomena, which is, in fact, the philosophical goal of
science, we will regard it as rash to aspire never, even for
the most distant future, to reduce them strictly to one.
In this discourse I have attempted to determine, as exactly as
was in my power, the aim, the spirit and the influence of
positive philosophy . So I marked the term towards which have always tended and
will constantly tend all my work, either in this course or in any
other way. No one is more deeply convinced than I of
the insufficiency of my intellectual forces, even if they are
much greater than their real value, to respond to such a
vast and lofty task . But what cannot be done by one
mind, nor in one life, only one can clearly propose. This
is my whole ambition.
Having explained the real aim of this course, that is to say fixed the point of
view from which I will consider the various principal branches of
natural philosophy, I will complete, in the next lesson, these
general prolegomena, passing to the exposition of the plan, that is to say to
the determination of the encyclopedic order that should be established
between the various classes of natural phenomena, and consequently
between the corresponding positive sciences .
SECOND LESSON.
SUMMARY. Exhibition of the outline of this course, or general considerations
on the hierarchy of positive sciences.
After having characterized as exactly as possible, in the
preceding lesson , the considerations to be presented in this course on all the
principal branches of natural philosophy, we must
now determine the plan which we must follow, that is to say, the
rational classification most suitable to establish between the
different fundamental positive sciences, in order to study them
successively from the point of view which we have fixed. This second
general discussion is essential to complete the understanding
from the outset of the true spirit of this course.
It is easy to see first of all that it is not a question here of making a
criticism, unfortunately too easy, of the numerous classifications
which have been proposed successively for two centuries, for the
general system of human knowledge, considered in all its
extent. We are now convinced that all scales
encyclopaedics constructed, like those of Bacon and d'Alembert,
according to any distinction between the various faculties of the
human mind , are by this alone radically vicious, even when this
distinction is not, as it often happens, more subtle than real;
for, in each of its spheres of activity, our understanding
simultaneously employs all its principal faculties. As for all the other
classifications proposed, it will suffice to observe that the various
discussions raised on this subject had the definitive result of showing
in each of the fundamental defects, so much that none could obtain
a unanimous agreement, and that in this respect there are almost as many
opinions than individuals. These various attempts have even been, in
general, so ill-conceived, that they have unwittingly resulted in the
majority of good minds in unfavorable prevention against any
enterprise of this kind.
Without dwelling more on a fact so well established, it is more
essential to seek its cause. However, one can easily explain the
deep imperfection of these encyclopedic attempts, so often
renewed until now. I need not point out that, since
the general discredit into which works of this nature have fallen as a
result of the lack of solidity of the first projects, these classifications
are most often only conceived by minds almost entirely
foreign to the knowledge of the objects to be classified. Regardless of
this personal consideration, there is a much more important one,
drawn from the very nature of the subject, and which clearly shows why
it has so far not been possible to amount to a
truly satisfactory encyclopedic conception . It consists in the lack
of homogeneity which has always existed until recently between the
different parts of the intellectual system, some having
successively become positive, while others remain
theological or metaphysical. In such an incoherent state of affairs,
it was obviously impossible to establish any
rational classification . How can we manage to have
such profoundly contradictory conceptions in a single system ? it is a difficulty
against which all
classifiers have necessarily failed , without any one having perceived it distinctly. It was very
evident, however, to anyone who was well acquainted with the true
state of the human mind, that such an enterprise was premature,
and that it could only be attempted with success when all our
main views had become positive.
This fundamental condition can now be regarded as
fulfilled, according to the explanations given in the previous lesson, it
is therefore possible to proceed to a truly rational
and lasting arrangement of a system in which all the parts have finally become
homogeneous.
On the other hand, the general theory of classifications, established in
recent times by the philosophical works of botanists and
zoologists, allows us to hope for real success in such work,
offering us a certain guide by the true fundamental principle.
of the art of classifying, which had never before been conceived distinctly
. This principle is a necessary consequence of the sole
direct application of the positive method to the very question of
classifications, which, like any other, must be treated by
observation, instead of being resolved by _a priori_ considerations.
It consists in that the classification must emerge from the very study
of the objects to be classified, and be determined by the real affinities and
the natural sequence that they present, so that this
classification is itself the expression of the the most general fact,
manifested by the in-depth comparison of the objects it embraces.
Applying this fundamental rule to the present case, it is therefore according to the
mutual dependence which actually takes place between the various
positive sciences that we must proceed to their classification; and this
dependence, to be real, can only result from that of
the corresponding phenomena.
But before carrying out, in such a spirit of observation, this
important encyclopedic operation, it is essential, in order not to
get lost in a too extensive work, to circumscribe with more
precision than we have done so far. , the proper subject of the
proposed classification.
All human work is, or speculation, or action. Thus, the
most general division of our real knowledge is to
distinguish them into theoretical and practical. If we first consider this
first division, it is obvious that it is only knowledge
theoretical that must be discussed in a course of the nature of
it; for, it is not a question of observing the entire system of
human notions, but only that of the fundamental conceptions
on the various orders of phenomena, which furnish a solid basis for
all our other combinations whatever, and which are not, to their turn,
based on no antecedent intellectual system. Now, in such a
work, it is speculation that must be considered, and not
the application, except in so far as the latter can clarify the
former. This is probably what Bacon understood, although very
imperfectly, by this _philosophy first_ which he indicates as
to be extracted from all the sciences, and which has been so
variously and always so strangely conceived by the metaphysicians who
have undertaken to comment on its thought.
Undoubtedly, when one considers the full body of work of all
kinds of the human species, one must conceive of the study of nature as
intended to provide the true rational basis for the action of man
on nature, since knowledge of the laws of phenomena, the
constant result of which is to make us foresee them, can only evidently
lead us, in active life, to modify them to our advantage
one by one. Our natural and direct means to act on
bodies around us are extremely weak, and
grossly out of proportion to our needs. Whenever we succeed in
exerting a great action, it is only because the knowledge of
natural laws enables us to introduce among the
determined circumstances under the influence of which the various
phenomena are accomplished , some modifying elements, which, however weak
in themselves they are sufficient, in certain cases, to turn to
our satisfaction the final results of all
external causes . In short, _science, hence foresight; foresight, hence
action_: such is the very simple formula which expresses, in a
exact, the general relation of _science_ and_art_, taking
these two expressions in their full meaning.
But, in spite of the capital importance of this relation, which must never
be ignored, it would be a very imperfect idea to form of the sciences
to conceive of them only as the bases of the arts, and this is
unfortunately what we are not doing. that too prone these days. Whatever might be
the immense service rendered to industry by
scientific theories , although, to use Bacon's energetic expression,
power is necessarily proportionate to knowledge, we
must not forget that the sciences have, above all, a destination.
more direct and higher, that of satisfying the fundamental need
felt by our intelligence to know the laws of phenomena. To
feel how deep and compelling this need is, it suffices to think for
a moment of the physiological effects of astonishment, and to consider
that the most terrible sensation that we can experience is the one
that occurs whenever a phenomenon occurs to us. seems to be accomplished in
contradiction to natural laws which are familiar to us. This
need to arrange the facts in an order which we can conceive
with ease (which is the specific object of all
scientific theories ) is so inherent in our organization, that, if we
If we did not manage to satisfy it with positive conceptions, we
would inevitably return to the theological and
metaphysical explanations to which it originally gave birth, as I
explained in the last lesson.
I thought I should expressly point out from this moment a consideration
which will recur frequently throughout the rest of this course, in order
to indicate the need to guard against the too great influence
of current habits which tend to prevent one from forming.
just and noble ideas of the importance and destination of science. If
the preponderant power of our organization did not correct, even
involuntarily, in the minds of the learned, what there is in this
respect of incomplete and narrow in the general tendency of our
time, the human intelligence, reduced to concern itself only with research
likely to an immediate practical utility would be found by that
alone, as Condorcet very rightly remarked, completely arrested
in its progress, even with regard to those applications to which
purely speculative work would have been imprudently sacrificed; for, the
most important applications constantly derive from theories
formed for a simple scientific intention, and which often have been
cultivated for several centuries without producing any result
convenient. We can cite a very remarkable example in the fine
speculations of the Greek geometers on the conical sections, which, after
a long succession of generations, served, by determining the renovation
of astronomy, to finally lead the art of navigation. to the degree
of perfection which he has attained in recent times, and which
he would never have attained without the purely theoretical works
of Archimedes and Apollonius; so much so that Condorcet was able to say with
reason in this regard: the sailor, whom an exact observation of
longitude preserves from shipwreck, owes his life to a theory conceived, two
thousand years before, by men of genius who had in view of
simple geometric speculations.
It is therefore evident that after having conceived, in a general way,
the study of nature as serving as a rational basis for action on
nature, the human mind must proceed to theoretical research,
completely disregarding any practical considerations; for,
our means for discovering the truth are so weak, that if we
do not concentrate them exclusively towards this goal, and if, by seeking
the truth, we impose on ourselves at the same time the foreign condition of
finding in it an immediate practical utility, it would almost always be
impossible for us to do that.
In any case, it is certain that all of our knowledge
on nature, and that of the processes which we deduce from it to
modify it to our advantage, form two systems essentially
distinct in themselves, which it is convenient to conceive and
cultivate separately. In addition, the first system being the basis of the
second, it is obviously the one that should be considered first in
a methodical study, even when one would propose to embrace the
totality of human knowledge, both application and of
speculation. This theoretical system seems to me to constitute
today exclusively the subject of a truly rational course in
positive philosophy: at least that is how I see it. Without
Doubtless, it would be possible to imagine a more extensive course, covering
both theoretical generalities and practical generalities.
But I do not think that such an enterprise, even regardless of its
scope, can be properly attempted in the present state of
the human mind. It seems to me, in fact, to require beforehand a
very important work and of a very particular nature, which has not
yet been done, that of forming, according to scientific theories
properly so called, the special conceptions intended to serve from
direct basics to general practice procedures.
To the degree of development already reached by our intelligence, it is not
not immediately that the sciences apply to the arts, at least in
the most perfect cases; there exists between these two orders of ideas an
average order, which, still badly determined in its philosophical character,
is already more noticeable when one considers the social class which is
specially concerned with it. Between the scholars properly so called and the
effective directors of productive work there is now beginning to form
an intermediate class, that of the _engineers_, whose
special purpose is to organize the relations between theory and practice.
Without having in any way the progress of scientific knowledge in view,
it considers them in their present state to deduce the
industrial applications of which they are susceptible. Such, at
least, is the natural tendency of things, although there is still
much confusion in this respect. The body of doctrine specific to this
new class , and which should constitute the true direct theories of the
various arts, could, without doubt, give rise to
philosophical considerations of great interest and of real importance. But a
work which embraces them jointly with those founded on the
sciences properly so called, would today be completely premature;
for, these doctrines intermediate between pure theory and
direct practice have not yet been formed: so far only
some imperfect elements relative to the most
advanced sciences and arts , and which only allow one to conceive of the nature and the
possibility of similar works for the whole of
human operations . It is thus, to cite the most important example of it here,
that we must consider the beautiful conception of Monge, relative to
descriptive geometry, which is really nothing other than a
general theory of the construction arts. I will take care to indicate
successively the small number of analogous ideas already formed, and to
show their importance as the natural development
of this course presents them to us. But it is clear that designs
Up to now also incomplete should not enter, as an
essential part , in a course of positive philosophy which should
include, as far as possible, only doctrines having a
fixed and clearly determined character.
The difficulty of constructing these
intermediate doctrines which I have just indicated will be all the better understood, if we consider that each art
depends not only on a certain corresponding science, but at the same
time on several, so much so that the arts the most important derive
direct aid from almost all the various principal sciences.
This is how the true theory of agriculture, to confine myself to
the most essential case, requires an intimate combination of
physiological, chemical, physical and even astronomical and
mathematical knowledge : it is the same with the fine arts. It is easy to see,
from this consideration, why these theories have not yet been
formed, since they suppose the prior development of all the
different fundamental sciences. It also results in a new
reason for not including such an order of ideas in a course in
positive philosophy, since, far from being able to contribute to the
systematic formation of this philosophy, the general theories specific to the
various principal arts must, on the contrary, , as we see,
to be later probably one of the most useful consequences of
its construction.
In summary, we should therefore consider in this course only
scientific theories and not their applications. But before proceeding to
the methodical classification of its different parts, it remains for me to
expose, relative to the sciences properly so called, an
important distinction , which will complete clearly circumscribing the proper subject of
the study which we are undertaking.
It is necessary to distinguish, in relation to all the orders of phenomena, two
kinds of natural sciences: the one abstract, general, have for
object the discovery of the laws which govern the various classes of
phenomena, considering all the cases that can be conceived; the others,
concrete, particular, descriptive, and which are sometimes designated
by the name of natural sciences properly so called, consist in
the application of these laws to the effective history of the various
existing beings . The first are therefore fundamental, it is
only on them that our studies will focus in this course; the others, whatever
their own importance, are really only secondary, and
must not, consequently, be part of a work which its
extreme natural extent obliges us to reduce to the smallest
possible development .
The preceding distinction cannot present any obscurity to the minds
who have some special knowledge of the different
positive sciences , since it is more or less the equivalent of that which is
ordinarily stated in almost all scientific treatises when comparing
dogmatic physics with natural history properly so called. A few
examples will suffice, moreover, to make this division visible,
the importance of which has not yet been properly appreciated.
We can first perceive it very clearly by comparing, on the one hand,
general physiology, and, on the other hand, zoology and
botany properly so called. It is evidently, in fact, two works
of a very distinct character, to study, in general, the laws of
life, or to determine the mode of existence of each living body, in
particular. This second study, moreover, is necessarily based
on the first.
It is the same with chemistry, with respect to mineralogy; the
first is obviously the rational basis of the second. In
chemistry, we consider all possible combinations of molecules,
and under all imaginable circumstances; in mineralogy, we
consider only those of these combinations which are realized
in the effective constitution of the terrestrial globe, and only under the influence
of the circumstances which are proper to it. This clearly shows
the difference from a chemical point of view and a mineralogical point of view,
although the two sciences bear on the same objects, it is because
most of the facts considered in the first have only an
artificial existence , so that a body, like chlorine or
potassium, can have an extreme importance in chemistry by the extent
and energy of its affinities, while it will have almost none in
mineralogy; and reciprocally, a compound, such as granite or
quartz, on which the greater part of
mineralogical considerations bear , will offer, under the chemical point of view, only
very mediocre interest .
This makes, in general, still more sensitive the logical necessity of
this fundamental distinction between the two great sections of the
natural philosophy is that not only does each section of
concrete physics presuppose the prior cultivation of the
corresponding section of abstract physics, but that it even requires
knowledge of the general laws relating to all orders of
phenomena. Thus, for example, not only does the special study of the
earth, considered from all the points of view that it can
effectively present , require a prior knowledge of physics and
chemistry, but it cannot be done properly without to introduce,
on the one hand, astronomical knowledge, and even, on the other hand,
physiological knowledge; so that it sticks to the system
whole of basic sciences. The same is true of each of
the natural sciences proper. It is precisely for this reason
that _ concrete physics_ has so far made so little
real progress , for it could not begin to be studied in a truly
rational way until after _abstract physics_, and when all the
various main branches of it have taken on their
final character , which has only taken place today. Until then we have only been able to
collect more or less incoherent material on this subject, which
is even still very incomplete. Known facts cannot be
coordinated in such a way as to form genuine special theories of
different beings of the universe, only when the fundamental distinction
recalled above, will be more deeply felt and more regularly
organized, and that, consequently, the scientists particularly devoted to
the study of the natural sciences properly so called, will have recognized the
necessity to base their research on an in-depth knowledge of
all the fundamental sciences, a condition which is still
far from being properly fulfilled today.
The examination of this condition clearly confirms why we must,
in this course of positive philosophy, reduce our considerations to
the study of the general sciences, without embracing at the same time the
descriptive or particular sciences. We see the birth here, in fact, of
a new essential property of this specific study of the generalities
of abstract physics; it is to provide the rational basis for a
truly systematic concrete physics. Thus, in the present state of
the human mind, there would be a sort of contradiction in wanting to
unite, in one and the same course, the two orders of sciences. We can
say, moreover, that even if concrete physics had already reached
the degree of perfection of abstract physics, and that,
consequently, it would be possible, in a course of positive philosophy,
to embrace at the same time the one and the other, it should not be less
obviously start with the abstract section, which will remain the
invariable basis of the other. It is clear, moreover, that the only study of the
generalities of the fundamental sciences is vast enough in itself,
for it to be important to set aside, as far as possible, all the
considerations which are not essential; now, those relating to
the secondary sciences will always be, whatever happens, of a
distinct kind . The philosophy of the fundamental sciences, presenting a
system of positive conceptions on all our orders of
real knowledge , suffices, by that very fact, to constitute this _philosophy first_
that Bacon sought, and which is destined to serve henceforth.
permanent basis for all human speculation, must be
carefully reduced to the simplest possible expression.
I do not need to dwell any further at this time on such a
discussion, which I will naturally have several opportunities to reproduce
in the various parts of this course. The preceding explanation is
sufficiently developed to justify the way in which I have circumscribed the
general subject of our considerations.
Thus, as a result of all that has just been exposed in this lesson,
we see: 1 ° that human science consisting, as a whole,
of speculative knowledge and knowledge of application, is
only the first ones that we have to deal with here; (2) that
theoretical knowledge or the sciences properly so called, being divided
into general sciences and particular sciences, we must
consider here only the first order, and limit ourselves to
abstract physics , whatever interest
concrete physics may present to us .
The proper subject of this course being thus precisely circumscribed, it is
now easy to proceed to a really
satisfactory rational classification of the fundamental sciences, which constitutes the
encyclopedic question , the special subject of this lesson.
Above all, we must begin by recognizing that, however natural
whatever
such a classification may be, it will always necessarily contain something, if not arbitrary, at least artificial,
so as to present a real imperfection.
Indeed, the main goal that we must have in view in any
encyclopedic work is to arrange the sciences in the order of their
natural sequence, following their mutual dependence; in such a
way that they can be exposed successively, without ever being drawn
into the slightest vicious circle. Now, this is a condition which it seems to me
impossible to fulfill in a completely rigorous manner. Allow me
to give some development to this reflection here,
I believe it is important to characterize the real difficulty of the
research which occupies us at present. This consideration, moreover,
will give me cause to establish, relative to the exposition of our
knowledge, a general principle of which I shall later have to present
frequent applications.
Any science can be exposed according to two essentially
distinct walks , of which any other mode of exposition can only be a
combination, the _historic_ walk, and the _dogmatic_ walk.
By the first process, we successively expound knowledge in
the same effective order in which the human mind actually
obtained it, and by adopting, as far as possible, the same paths.
By the second, we present the system of ideas as it could be
conceived today by a single mind, which, placed in the
proper point of view , and provided with sufficient knowledge, would occupy itself in
remaking science as a whole. .
The first mode is obviously that by which begins, of all
necessity, the study of each emerging science; because, it presents this
property, of not requiring, for the exhibition of knowledge, any
new work distinct from that of their formation, all the didactics
being reduced then to study successively, in
chronological order , the various original works which have contributed to the
advancement of science.
The dogmatic mode, supposing on the contrary that all these
particular works have been recast into a general system, to be presented
in a more natural logical order, is applicable only to a science
already reached a rather high degree of development. But, as
science advances, the _historical_ order of exposition becomes
more and more impracticable, by the too long series of intermediaries which
it would oblige the mind to go through; while the _dogmatic_ order
becomes more and more possible, at the same time as it is necessary, because
new conceptions make it possible to present the
earlier discoveries under a more direct point of view.
It is thus, for example, that the education of a geometer of antiquity
consisted simply in the successive study of the very small number of
original treatises produced until then on the various parts of
geometry, which essentially reduced to the writings of Archimedes and
Apollonius; while, on the contrary, a modern geometer has
commonly completed his education without having read a single
original work , except in relation to the most recent discoveries, which
can only be known by this means.
The constant tendency of the human mind, as regards the exhibition of
knowledge, is therefore to substitute more and more for the
historical order the dogmatic order,
perfected our intelligence.
The general problem of intellectual education consists in bringing
, in a few years, a single understanding, usually
mediocre, to the same point of development which has been attained, in a
long series of centuries, by a great number of superior geniuses
successively applying, during their whole life, all their forces
to the study of the same subject. It is clear, from this, that, although it
is infinitely easier and shorter to learn than to invent, it
would certainly be impossible to achieve the proposed goal, if one were
to subject each individual mind to pass successively through
the same intermediaries that the
collective genius of the human species must necessarily have followed . Hence the indispensable need for a
dogmatic order , which is especially so sensitive today for
the most advanced sciences , whose ordinary mode of exposition no longer presents
almost any trace of the effective filiation of their details.
It should, however, be added, in order to prevent any exaggeration, that any
real mode of exposition is, inevitably, some combination of
the dogmatic order with the historical order, in which only the
former must constantly and more and more dominate. . The dogmatic order
cannot, in fact, be followed in a completely rigorous manner; because,
by the very fact that it requires a new elaboration of
acquired knowledge , it is not applicable, in every epoch of science, to the
recently formed parts, the study of which comprises only an
essentially historical order , which does not present , moreover, in this
case, the main drawbacks which make it generally rejected.
The only fundamental imperfection which one could reproach the
dogmatic mode is to allow ignoring the manner in which
various human knowledge was formed , which, although distinct from
the actual acquisition of this knowledge, is, in itself, of the highest
interest in any philosophical mind. This consideration would, to
my eyes, a lot of weight, if it was really a motive in favor
of the historical order. But it is easy to see that there is only
an apparent relation between studying a science by following the so-called
_historical_ mode , and really knowing the actual history of this
science.
Indeed, not only the various parts of each science, which we
are led to separate in the _dogmatic_ order, have in reality
developed simultaneously and under the influence of one another,
which would tend to favor the _historical_ order: but considering,
as a whole, the effective development of the human mind, we further see
that the different sciences have been,
improved at the same time and mutually; we even see that the
progress of the sciences and those of the arts have depended on one another,
by innumerable reciprocal influences, and finally that all have been
closely linked to the general development of human society. This
vast concatenation is so real that often, in order to conceive of the
effective generation of a scientific theory, the mind is led to
consider the improvement of some art which has no
rational connection with it , or even some particular progress in it
. social organization, without which this discovery could not have taken
place. We will see many examples of this below. It results
therefore from there that one cannot know the true history of each
science, that is to say, the real formation of the discoveries of which it is
composed, only by studying, in a general and direct manner, the history
of mankind. That is why all the documents collected so far
on the history of mathematics, astronomy, medicine, etc.,
however precious they may be, can only be regarded as
material.
The so-called _historical_ order of exposition, even when it could be
followed rigorously for the details of each particular science,
would already be purely hypothetical and abstract in the most important respect.
important, in that he would consider the development of this science to
be isolated. Far from bringing out the true history of
science, it would tend to give rise to a very false opinion of it.
Thus, we are certainly convinced that knowledge of
the history of science is of the utmost importance. I even think that
we do not know a science completely until we know
its history. But this study must be conceived as entirely separate
from the proper and dogmatic study of science, without which even this
history would not be intelligible. We will therefore consider with
great care the real history of the fundamental sciences which go
to be the subject of our meditations; but it will only be in the
last part of this course, that relating to the study of
social phenomena , dealing with the general development of humanity, of which
the history of the sciences constitutes the most important part, although
so far the most important part. more neglected. In the study of each science, the
incidental historical considerations which may arise will have
a clearly distinct character, so as not to alter the
proper nature of our main work.
The preceding discussion, which must, moreover, as we see, be
specially developed later, tends to clarify further, by
presenting from a new point of view, the true spirit of this
course. But above all, the result, relative to the
current question , is the exact determination of the conditions that must be imposed and
that one can justly hope to fulfill in the construction of an
encyclopedic scale of the various fundamental sciences.
We see, in fact, that, however perfect it may be supposed, this
classification can never be strictly in accordance with
the historical sequence of the sciences. Whatever one does, one can
not entirely avoid presenting as anterior a
certain science which will nevertheless need, in some particular respects more or less
important, to borrow concepts from another science classified in a
later rank. It is only necessary to try that such an inconvenience does not
occur in relation to the characteristic conceptions of each science,
because then the classification would be altogether vicious.
Thus, for example, it seems to me indisputable that, in the
general system of sciences, astronomy must be placed before physics
proper, and nevertheless several branches of the latter, especially
optics, are indispensable for the exposition full of the first.
Such secondary defects, which are strictly inevitable, can
not prevail against a classification, which would moreover fulfill
suitably the main conditions. They are due to what is
necessarily artificial in our division of intellectual labor.
Nevertheless, although, according to the preceding explanations, we
should not take the historical order as the basis of our
classification, I should not neglect to indicate in advance, as an
essential property of the encyclopedic scale which I will propose,
its general conformity with the whole of scientific history; in the
sense that, despite the real and continuous simultaneity of the development
of the different sciences, those which will be classified as earlier
will, in fact, be older and constantly more advanced than those
presented as posterior. This is what must
inevitably take place if, in reality, we take, as it should be, as a
principle of classification, the natural logical sequence of the various
sciences, the starting point of the species having necessarily had to be the
same. than that of the individual.
In order to complete the determination with all possible precision of the
exact difficulty of the encyclopedic question which we have to
resolve, I believe it useful to introduce a
very simple mathematical consideration which will rigorously summarize the whole of the reasonings
exposed so far in this lesson. Here is what it consists of.
We propose to classify the basic sciences. Now we
we will soon see that, all things considered, it is not possible to
distinguish less than six; most scholars would
probably even admit a larger number. That said, we know that six
objects have 720 different arrangements. The
basic sciences could therefore give rise to 720
distinct classifications , from which it is a question of choosing the
necessarily unique classification which best satisfies the main conditions
of the problem. It can be seen that, despite the large number of
encyclopedic scales successively proposed so far, the discussion
has only focused on a very small part of the provisions.
possible; and nevertheless, I think I can say without exaggeration that in
examining each of these 720 classifications, there might
not be a single one in favor of which some
plausible reasons could not be advanced; for, on observing the various arrangements which have
been effectively proposed, one notices the most extreme
differences between them ; the sciences which are placed by some at the head of
the encyclopedic system, being referred by others to the
opposite end , and vice versa. It is therefore in this choice of a single
truly rational order , among the very considerable number of
possible systems , that the precise difficulty of the question which we
have posed consists .
Approaching now in a direct way this great question,
let us first remember that in order to obtain a natural and
positive classification of the fundamental sciences, it is in the comparison of the
various orders of phenomena of which their object is to discover the
laws that we must seek the principle. What we want to
determine is the actual dependence of the various
scientific studies . Now, this dependence can only result from that of
the corresponding phenomena.
Considering all observable phenomena from this point of view,
we will see that it is possible to classify them into a small number of
natural categories, arranged in such a way that the study
rationality of each category is based on knowledge of the
main laws of the preceding category, and becomes the basis for
the study of the next. This order is determined by the degree of
simplicity, or, which amounts to the same thing, by the degree of generality of the
phenomena, from which results their successive dependence, and, consequently,
the greater or lesser facility of their study.
It is in fact clear, _a priori_, that the simplest phenomena,
those which are least complicated by others, are necessarily also
the most general; for, what is observed in the greatest number of
cases is, by that very fact, freed as much as possible from the circumstances
specific to each separate case. It is therefore with the study of the
most general or the simplest phenomena that it is necessary to begin, by proceeding
then successively to the most particular or the
most complicated phenomena , if one wishes to conceive the natural philosophy of a
really methodical way; for, this order of generality or of
simplicity necessarily determining the rational sequence of the
various fundamental sciences by the successive dependence of their
phenomena, thus fixes their degree of ease.
At the same time, by an auxiliary consideration which I believe important
to note here, and which converges exactly with all the preceding ones,
the most general or the simplest phenomena
necessarily being the most foreign to man, must, by that very fact,
be studied in a calmer, more rational frame of mind,
which constitutes a new motive for the sciences corresponding ones
develop faster.
Having thus indicated the fundamental rule which must govern the
classification of the sciences, I can proceed immediately to the
construction of the encyclopedic scale according to which the outline of this
course must be determined, and which each one can easily assess with
the help of of the preceding considerations.
A first contemplation of all natural phenomena
The first step is to divide them, in accordance with the principle which we have
just established, into two great main classes, the first comprising
all the phenomena of gross bodies, the second all those of
organized bodies .
The latter are obviously, in fact, more complicated and more
particular than the others; they depend on the precedents, which, on the
contrary, do not depend on them at all. Hence the need to study
physiological phenomena only after those of inorganic bodies. However
we explain the differences between these two kinds
of beings, it is certain that we observe in living bodies all the
phenomena, whether mechanical or chemical, which take place in the body.
crude, plus a very special order of phenomena, the vital phenomena
properly so called, those which depend on the_organization_. It is not a question
here of examining whether the two classes of bodies are or are not of the
same _nature_, an insoluble question which is still agitated much too much
nowadays, by a remainder of influence of theological and
metaphysical habits ; such a question does not
fall within the domain of
positive philosophy , which formally professes to ignore absolutely _the intimate nature_ of any body whatever. But it is by no means
essential to consider gross bodies and living bodies as
being of an essentially different nature in order to recognize the
need to separate their studies.
No doubt, ideas are not yet sufficiently fixed on the
general manner of conceiving the phenomena of living bodies. But,
whatever part one may take in this regard as a result of the
subsequent progress of natural philosophy, the classification which we
establish cannot be affected in any way. Indeed, if
we looked at it as demonstrated, what barely allows us to glimpse the
present state of physiology, that physiological phenomena are
always simple mechanical, electrical and chemical phenomena,
modified by the structure and composition specific to bodies. organized,
our fundamental division would still exist. For it
always remains true, even in this hypothesis, that general phenomena
must be studied before proceeding to the examination of the
special modifications which they experience in certain beings of the universe, as a result
of a particular arrangement of the molecules. Thus, the division, which
today is founded in most enlightened minds on the
diversity of laws, is of a nature to be maintained indefinitely because of
the subordination of phenomena and as a result of studies, however
close one can ever be. establish firmly between the two
body classes.
This is not the place to develop, in its various parts
essentielles, la comparaison générale entre les corps bruts et les corps
vivans, which will be the special subject of an in-depth examination in the
physiological section of this course. It suffices, for the present, to have recognized,
in principle, the logical necessity of separating the science relating to the
former from that relating to the latter, and to proceed to the study of
_organic physics_ only after having established the laws. generalities of
_inorganic physics_.
Let us now pass to the determination of the principal sub-division of
which each of these two great
halves of natural philosophy is susceptible, according to the same rule .
For _inorganic physics_, we first see, by
always conforming to the order of generality and dependence of phenomena,
that it should be divided into two distinct sections, according to whether it
considers the general phenomena of the universe, or, in particular, those
presented by terrestrial bodies. Hence celestial physics, or
astronomy, either geometrical or mechanical; and
terrestrial physics . The need for this division is exactly the same
as for the previous one.
Astronomical phenomena being the most general, the simplest,
the most abstract of all, it is obviously with their study that
natural philosophy must begin, since the laws to which they are
subject influence those of all other phenomena, including
they themselves are, on the contrary, essentially independent. In all
the phenomena of terrestrial physics, we first observe the
general effects of universal gravitation, plus some other effects which
are specific to them, and which modify the former. It follows that
when we analyze the simplest terrestrial phenomenon, not only
by taking a chemical phenomenon, but by choosing even a
purely mechanical phenomenon , we find it constantly more compound than
the most complicated celestial phenomenon. It is thus, for example, that the
simple movement of a heavy body, even when it is only a question of a
solid, really presents, when we want to take into account all the
determining circumstances, a more complicated research subject than
the most difficult astronomical question. Such a consideration
clearly shows how essential it is to clearly separate
celestial physics and terrestrial physics, and to proceed to the study
of the second only after that of the first, which is its
rational basis .
Terrestrial physics, in its turn, is subdivided, according to the same
principle, into two very distinct portions, according to whether it considers
bodies from the mechanical point of view, or from the chemical point of view.
Hence physics proper, and chemistry. This, to be
conceived in a truly methodical way, obviously presupposes the
prior knowledge of the other. For all chemical phenomena
are necessarily more complicated than physical phenomena; they
depend on it without influencing them. Everyone knows, in fact, that all
chemical action is first subjected to the influence of gravity,
heat, electricity, etc., and presents, in addition, something of its
own which modifies the action. of the preceding agents. This consideration,
which obviously shows chemistry as being able to work only after
physics, presents it at the same time as a distinct science. Because,
whatever opinion one adopts relative to chemical affinities, and
even so one would see in them, as one can conceive it, only
from the modifications of the general gravitation produced by the figure and
by the mutual arrangement of the atoms, it would remain incontestable that
the necessity to have continual consideration of these special conditions
would not allow chemistry to be treated as a simple appendix of
physics. We would therefore be obliged, in all cases, were it only for
the ease of the study, to maintain the division and the concatenation which
we regard today as due to the heterogeneity of phenomena.
Such, then, is the rational distribution of the principal branches of
the general science of gross bodies. An analogous division is established in
the same way in the general science of organized bodies.
All living beings present two
essentially distinct orders of phenomena , those relating to the individual, and those which
concern the species, especially when it is sociable. It is
mainly in relation to man that this distinction is
fundamental. The last order of phenomena is obviously more
complicated and more particular than the first; it depends
on it without influencing it. Hence two major sections in _organic physics_,
physiology proper, and social physics, which is based on
the first.
In all social phenomena we first observe the influence of
the physiological laws of the individual, and, moreover, something
particular which modifies its effects, and which is due to the action of
individuals on each other, singularly complicated, in
the human species, by the action of each generation on the one that
follows it. It is therefore evident that, in order to study
social phenomena properly , one must first start with a
thorough knowledge of the laws relating to individual life. On the other hand,
this necessary subordination between the two studies in no
way prescribes , as some first-class physiologists have been
led to believe, to see social physics as a mere appendage of
physiology. Although the phenomena are certainly homogeneous, they
are not identical, and the separation of the two sciences is of
really fundamental importance. For, it would be impossible to treat
the collective study of the species as a pure deduction from the study of
the individual, since the social conditions, which modify the action of
physiological laws, are precisely then the most
essential consideration. . Thus, social physics must be founded on a body
of direct observations which is proper to it, while having due regard, as
appropriate, to its necessary intimate relation with physiology
proper.
inorganic, recalling the vulgar distinction of physiology
proper in plant and animal. It would be easy, in fact, to
relate this sub-division to the principle of classification which we
have constantly followed, since the phenomena of animal life
present themselves, in general at least, as more complicated and more special
than those of vegetable life. . But there
would be something childish about the search for this precise symmetry if it led to ignoring or
exaggerating the real analogies or the effective differences of the
phenomena. Now, it is certain that the distinction between
plant physiology and animal physiology, which is of great importance in this
which I have called _physics concrete_, has almost none in
_physics abstract_, the only one which is concerned here. The knowledge of
the general laws of life, which must be, in our eyes, the true object
of physiology, requires the simultaneous consideration of the whole
organic series without distinction of plants and animals, a distinction which,
moreover, is essential. 'clears day by day, as the phenomena are
studied in more depth.
We will therefore persist in considering only one division in
organic physics, although we have thought it necessary to establish two
successive ones in inorganic physics.
As a result of this discussion,
naturally divided into five fundamental sciences, the
succession of which is determined by a necessary and
invariable subordination , founded, independently of any hypothetical opinion, on the
simple in-depth comparison of the corresponding phenomena: these are
astronomy, physics, chemistry, physiology , and finally
social physics. The first considers the most
general, the simplest, the most abstract and the most distant phenomena of
humanity; they influence all the others, without being influenced by
them. The phenomena considered by the latter are, on the contrary, the
most particular, the most complicated, the most concrete and the most
directly interesting for humans; they depend, more or less, on
all the precedents, without exercising any influence over them. Between these
two extremes, the degrees of specialty, complication, and
personality of the phenomena gradually increase, as does
their successive dependence. Such is the intimate general relation which
true philosophical observation, properly employed, and not
vain arbitrary distinctions, leads us to establish between the
various fundamental sciences. This must therefore be the plan for this course.
I have only been able here to outline the exposition of the main considerations
on which this classification is based. To design it
completely, it would now be necessary, after having considered it from a
general point of view, to examine it in relation to each fundamental science in
particular. We will do this carefully as we begin
the special study of each part of this course. The construction of this
encyclopedic scale, thus taken up successively, starting from
each of the five great sciences, will make it acquire more accuracy,
and above all will fully demonstrate its solidity. These advantages
will be all the more noticeable as we will then see the
internal distribution of each science naturally established according to the same
principle, which will present the whole system of human knowledge.
decomposed, down to its secondary details, according to a
single consideration constantly followed, that of the
greater or lesser degree of abstraction of the corresponding conceptions. But work of
this kind, apart from taking us far too
far now, would certainly be out of place in this lesson, where our mind
must maintain itself at the most general point of view of
positive philosophy .
Nevertheless, in order to appreciate as completely as possible, from this
moment, the importance of this fundamental hierarchy, of which I will make
continual applications throughout the rest of this course, I must
quickly point out here its most essential general properties.
It should first be noted, as a very decisive verification of
the accuracy of this classification, its essential conformity with the
co-ordination, in a way spontaneous, which is in fact
implicitly admitted by the scholars devoted to the study of the various
branches of natural philosophy.
It is a condition usually very neglected by the builders
of encyclopedic scales, to present as distinct the
sciences which the effective march of the human mind has led, without
premeditated design, to cultivate separately, and to establish among them a
subordination consistent with the positive relationships manifested by their
daily development. Such an agreement is nevertheless obviously the
surest indication of a good classification; for the divisions which
spontaneously introduced themselves into the scientific system could only be
determined by a long-felt sense of the true
needs of the human mind, without having been led astray by
vicious generalities.
But, although the classification proposed above entirely fulfills
this condition, which it would be superfluous to prove, it should
not be concluded that the habits generally established today by
experience among scholars, would render encyclopedic work unnecessary.
that we have just executed. They only made possible such an
operation, which presents the fundamental difference of a
rational conception to a purely empirical classification. It takes
besides that this classification is usually designed especially
followed with the necessary precision, and its importance is
properly appreciated; it would suffice, to be convinced of this, to
consider the serious offenses which are committed every day
against this encyclopedic law, to the great prejudice of the human spirit.
A second very essential characteristic of our classification is that it
necessarily conforms to the effective order of the development of the
natural philosophy. This is verified by everything we know about
the history of science, particularly in the last two
centuries, where we can follow their progress with more accuracy.
We can see, in fact, that the rational study of each
fundamental science requiring the prior culture of all those which
precede it in our encyclopedic hierarchy, could not make
real progress and take on its true character, only after a great
development. previous sciences relating to more
general, more abstract, less complicated phenomena , and independent of others.
It is therefore in this order that the progression, although simultaneous, must
have taken place.
This consideration seems to me of such importance that I do
not believe it is possible to really understand, without having regard to it, the history
of the human mind. The general law which dominates all this history, and
which I exposed in the preceding lesson, cannot be properly
understood, if one does not combine it in application with the
encyclopedic formula which we have just established. Because, it is according to the order
stated by this formula that the different human theories have
successively reached, first the theological state, then the
metaphysical state , and finally the positive state. If we do not take into account in
the use of the law this necessary progression,
often difficulties which will seem insurmountable, for it is clear
that the theological or metaphysical state of certain
fundamental theories must have temporarily coincided and sometimes did
indeed coincide with the positive state of those which predate them in our
encyclopedic system, which tends to throw
an obscurity over the verification of the general law which can only be dispelled by the
preceding classification.
In the third place, this classification presents the
very remarkable property of marking exactly the relative perfection of the
different sciences, which consists essentially in the degree of
precision of knowledge, and in their more or less co-ordination.
respondent.
It is easy to feel, in fact, that the more general,
simple and abstract the phenomena , the less they depend on others, and the more
precise the knowledge which relates to them can be, at the same time
as their coordination can be more complete. Thus,
organic phenomena only include a study which is both less exact and less
systematic than the phenomena of crude bodies. Likewise, in
inorganic physics, celestial phenomena, given their greater
generality and their independence from all others, have given rise to a
science much more precise and much more closely linked than that of
terrestrial phenomena .
This observation, which is so striking in the effective study of the
sciences, and which has often given rise to chimerical hopes or
unfair comparisons, is therefore completely explained by
the encyclopedic order which I have established. I will naturally have the opportunity
to give it its full extension in the next lesson, by showing
that the possibility of applying
mathematical analysis to the study of various phenomena , which is the means of obtaining for this study the highest
degree possible of precision and coordination, is exactly
determined by the rank that these phenomena occupy in my
encyclopedic scale .
I must not pass to another consideration,
The reader warns on this subject against a very serious error, and which, although
very gross, is still extremely common. It consists in
confusing the degree of precision which our different
knowledge entails with their degree of certainty, whence has resulted the
very dangerous prejudice that, the first being evidently very unequal, it must
be so with the second. So we often still speak, although less than in the
past, of the unequal certainty of the various sciences, which tends
directly to discourage the cultivation of the most difficult sciences. It
is clear, however, that precision and certainty are two
very different qualities in themselves. Quite a proposition
absurd can be extremely precise, as if we said, for
example, that the sum of the angles of a triangle is equal to three right angles
; and a very certain proposition may contain only
very mediocre precision, as when one affirms, for example, that every
man will die. If, according to the preceding explanation, the various
sciences must necessarily present a very unequal precision, it
is not at all so with their certainty. Each can offer
results as certain as those of any other, provided that it knows how to include
its conclusions in the degree of precision which the
corresponding phenomena entail , a condition which may not always be
very easy to fill. In any science whatever, everything which is
simply conjectural is only more or less probable, and this is not
what composes its essential domain; all that is positive,
that is to say, founded on well-established facts, is certain: there is
no distinction in this respect.
Finally, the most interesting property of our
encyclopedic formula , because of the importance and the multiplicity of
immediate applications that can be made of it, is to
directly determine the true general plan of an
entirely rational scientific education . This is what immediately results from the
composition of the formula.
It is, in fact, sensible that before undertaking the methodical study of
any of the fundamental sciences, it is necessary to have
prepared oneself by examining those relating to previous phenomena in
our encyclopedic scale, since these these always have a
preponderant influence on those whose laws one proposes to know.
This consideration is so striking that, in spite of its extreme
practical importance, I need not insist more at this
moment on a principle which, later, will
inevitably be reproduced in relation to each fundamental science. I will
confine myself only to observing that, while it is eminently applicable
in general education, it is also particularly important in the
special education of savants.
Thus, physicists who did not first study astronomy, at
least from a general point of view; chemists who, before
dealing with their own science, have not previously studied
astronomy and then physics; physiologists who have
not prepared themselves for their special works by a preliminary study of
astronomy, physics and chemistry have failed in one of the
fundamental conditions of their intellectual development. It is
even more evidently the same for minds who want to engage in
the positive study of social phenomena,
general knowledge of astronomy, physics, chemistry and
physiology.
As such conditions are very seldom fulfilled nowadays, and
no regular institution is organized to fulfill them, we
can say that there is no
really rational education yet for scholars . This consideration is, in my eyes, of such
great importance, that I do not fear to attribute in part to this defect
of our present education, the state of extreme imperfection in which we
still see the most difficult sciences, state truly
inferior to that prescribed by the more complicated nature of
the corresponding phenomena.
With respect to general education, this condition is even
more necessary. I believe it to be so indispensable that I regard
scientific education as incapable of achieving the
most essential general results which it is intended to produce in
society for the renewal of the intellectual system, if the various
principal branches of natural philosophy do not are not studied
in the correct order. Let us not forget that in almost all
intelligences, even the highest, ideas are usually
linked according to the order of their first acquisition; and that,
consequently, it is usually an irremediable evil not to have
started from the beginning. Each century has only a very small
number of thinkers capable, at the time of their virility, like Bacon,
Descartes and Leïbnitz, of making a real clean sweep, in order to
rebuild from top to bottom the entire system of their
acquired ideas .
The importance of our encyclopedic law to serve as a basis for
scientific education can only be properly appreciated by
considering it also in relation to the method, instead of
only considering it , as we have just done, relatively to doctrine.
From this new point of view, a suitable execution of the general plan
of studies which we have determined must have the necessary result of
to provide us with a perfect knowledge of the positive method, which
could not be obtained in any other way.
Indeed, the natural phenomena having been classified in such a way that
those which are really homogeneous always remain included in the
same study, while those which have been assigned to
different studies are effectively heterogeneous, it must necessarily
follow that the method general positive will be constantly modified
in a uniform manner within the scope of the same fundamental science,
and that it will constantly experience different and more
and more compound modifications , as it passes from one science to another. We will have
thus thus the certainty of considering it in all the
real varieties of which it is susceptible, which could not have taken place if
we had adopted an encyclopedic formula which does not
meet the essential conditions set out above.
This new consideration is of really fundamental importance;
for, if we saw in general, in the last lesson, that it is
impossible to know the positive method, when we want to study it
separately from its use, we must add today that we can
not to form a clear and exact idea by studying successively, and
in the proper order, its application to all the various classes
main of natural phenomena. A single science would not suffice
to achieve this goal, even by choosing it as
judiciously as possible. For, although the method is essentially
identical in all, each science specially develops one or another
of its characteristic procedures, the influence of which, too little pronounced
in the other sciences, would remain unnoticed. Thus, for example,
in certain branches of philosophy it is observation
properly so called; in others it is experience, and this or that
kind of experience, that is the main means of exploration. Similarly
, a certain general precept, which is an integral part of the method, has
was originally provided by a certain science; and although it may
then have been transported to others, it is at its source that it must be
studied in order to know it well; like, for example, the theory of
classifications.
By limiting oneself to the study of a single science, it would probably
be necessary to choose the most perfect, in order to have a more profound feeling of the
positive method. Now, the most perfect being at the same time the
simplest, we would thus have only a very incomplete knowledge of the
method, since we would not learn what essential modifications
it must undergo in order to adapt to more complicated phenomena. Each
fundamental science therefore has its
own advantages in this respect ; which clearly proves the need to consider them all,
under penalty of forming too narrow conceptions and
insufficient habits. As this consideration should recur
frequently in the following, it is unnecessary to develop it further at
this time.
I must nevertheless here, always with regard to method, insist
especially on the need, in order to know it well, not only
to study philosophically all the various fundamental sciences,
but to study them according to the encyclopedic order established in this
lesson. What can be rational, unless there is an extreme
natural superiority , a mind which deals first of all with the study of
the most complicated phenomena, without having previously learned to
know, by examining the simplest phenomena? , what is
a _law_, what is_observer_, what is a
positive conception , what is even reasoning followed? Such is, however,
still today the ordinary course of our young physiologists,
who immediately approach the study of living bodies, without having
generally been prepared other than by a preliminary education
reduced to the study of one or two dead languages. , and not having,
more than a very superficial knowledge of physics and
chemistry, a knowledge almost nil as regards method,
since it has not been commonly obtained in a rational manner,
and starting from the true point of departure from natural philosophy.
We can see how important it is to reform such a
vicious study plan . Likewise, with regard to social phenomena, which are even
more complicated, would it not have taken a great step towards the
return of modern societies to a truly normal state, than to have
recognized the logical necessity of not proceeding to the study of these phenomena,
only after having successively trained the intellectual organ by
philosophical depth of all previous phenomena? We can even
say with precision that this is the main difficulty. For there
are few good minds who are not convinced today that it is necessary to
study social phenomena according to the positive method. Only,
those who take care of this study, not knowing and not being able to
know exactly what this method consists of, for lack of having
examined it in its previous applications, this maxim has so
far remained sterile for the renovation of social theories, which
have not yet emerged from the theological or the
metaphysical state , despite the efforts of the so-called positive reformers.
This consideration will be specially developed later; I must
limit myself here to indicating it, only to show the full
scope of the encyclopedic conception that I have proposed in this
lesson.
These are therefore the four main points of view, from which I
have had to endeavor to emphasize the general importance of the
rational and positive classification established above for the
basic sciences.
In order to complete the general exposition of the outline of this course, it remains for me
now to consider an immense and capital gap, which I left
on purpose in my encyclopedic formula, and which the reader has no doubt
already noticed. In fact, in our
scientific system we have not marked the rank of mathematical science.
The reason for this voluntary omission is in the very importance of
this science, so vast and so fundamental. Because, the next lesson
will be entirely devoted to the exact determination of its true
general character, and consequently to the precise fixing of its
encyclopedic rank . But in order not to leave incomplete, in such an
essential respect , the large picture that I have tried to sketch in this
lesson, I must indicate here briefly, in anticipation, the
general results of the examination which we will undertake in the lesson.
next.
In the present state of the development of our positive knowledge, it is
appropriate, I believe, to regard mathematical science less as a
constituent part of natural philosophy proper, than
as being, since Descartes and Newton, the true fundamental basis of
all this philosophy, although, to speak exactly, it is
both one and the other. Today, in fact, mathematical science
is much less important for the knowledge, very real and
nevertheless very precious, which directly composes it, than as
constituting the most powerful instrument which the human mind can
employ in knowledge. research of the laws of natural phenomena.
In order to present a perfectly clear and
rigorously exact conception in this respect , we will see that it is necessary to divide
mathematical science into two great sciences, the character of which is
essentially distinct: abstract mathematics, or _calcul_, by
taking this word in its most great extension, and
concrete mathematics , which is composed, on the one hand of general geometry, on the
other hand of rational mechanics. The concrete part is
necessarily founded on the abstract part, and in turn becomes the
direct basis of all natural philosophy, considering, as far
as possible, all the phenomena of the universe as geometric or
mechanical.
The abstract part is the only one which is purely instrumental,
being nothing more than an immense and admirable extension of
natural logic to a certain order of deductions. Geometry and mechanics
must, on the contrary, be considered as true
natural sciences , founded like all the others, on observation,
although, by the extreme simplicity of their phenomena, they include
an infinitely more perfect degree of systematization. , which has
sometimes been able to misunderstand the experimental character of their
first principles. But these two physical sciences have this
peculiarity, that, in the present state of the human mind, they are
already and will always be used more as a method, much more
than as a direct doctrine.
It is, moreover, evident that by placing mathematical science at
the head of positive philosophy in this way, we are only extending further
the application of this same principle of classification, founded on the
successive dependence of the sciences as a result of degree of abstraction of
their respective phenomena, which has provided us with the encyclopedic series,
established in this lesson. We are now only returning
this series to its true first term, the importance of which
required further special examination. We see, in fact, that
Geometric and mechanical phenomena are, of all, the most general,
the simplest, the most abstract, the most irreducible, and the
most independent of all the others, of which they are, on the contrary, the
basis. We can also see that their study is an
indispensable preliminary to that of all the other orders of phenomena. It is
therefore mathematical science which must constitute the real
starting point of any rational scientific education, whether general or
special, which explains the universal use which has
long been established on this subject, of a empirically, although it
originally had no other cause than the greater relative antiquity of the
mathematical science. I must confine myself at this moment to a
very rapid indication of these various considerations, which will be the
special object of the following lesson.
We have therefore determined exactly in this lesson, not according to
vain arbitrary speculations, but by considering it as the subject
of a real philosophical problem, the rational plan which must
constantly guide us in the study of positive philosophy. As a
final result , mathematics, astronomy, physics, chemistry,
physiology, and social physics; such is the encyclopedic formula
which, among the very large number of classifications contained in
six fundamental sciences, alone is logically consistent with the
natural and invariable hierarchy of phenomena. I do not need to
reiterate the importance of this result, which the reader must become
eminently familiar with, in order to make it
continually throughout the scope of this course .
The final consequence of this lesson, expressed in the
simplest form, therefore consists in the explanation and justification of the large
synoptic table placed at the beginning of this work, and in the
construction of which I have endeavored to follow, as rigorously
as possible, for the internal distribution of each
fundamental science , the same principle of classification that comes from us
provide the general series of sciences.
THIRD LESSON.
SUMMARY. Philosophical considerations on the whole of
mathematical science .
Beginning to enter directly into the material by the philosophical study
of the first of the six fundamental sciences established in the
preceding lesson , we immediately see the importance of
positive philosophy in perfecting the general character of
each science in particular.
Although mathematical science is the oldest and most
perfect of all, the general idea which should be formed of it is not
yet clearly determined. The definition of science, its
main divisions, have so far remained vague and uncertain.
The multiple name by which it is usually designated would suffice even
alone to indicate the lack of unity of its philosophical character,
as it is commonly conceived.
Indeed, it was only at the beginning of the last century that the
various fundamental conceptions which constitute this great science
each took on sufficient development that the true spirit of
the whole could manifest itself clearly. Since that time,
the attention of surveyors has been too just and too exclusively
absorbed by the special improvement of the different branches, and
by the capital application which they have made of it to the most
important laws of the universe, in order to be able to direct themselves appropriately on the
general system of science.
But today the progress of specialties is no longer so rapid
that it prohibits contemplation of the whole. Mathematics [2] is
now sufficiently developed, either in itself or in its
most essential applications, to have reached this state of
consistency, in which one must strive to coordinate
the various parts in a single system. of science, in order to prepare for further
progress. We can even observe that the last improvements
capitals tested by mathematical science directly prepared
this important philosophical operation, by imparting to its main
parts a character of unity which did not exist before; such is
eminently and beyond all comparison the spirit of the works of
the immortal author of the _Théorie des Functions_ and of _Mechanics
analytic_.
[Note 2: I will often use this expression in the
singular, as Condorcet proposed, in order to indicate with
more energy the spirit of unity in which I conceive
science.]
To form a fair idea of the object of mathematical science
considered as a whole, we can
vague and insignificant that we usually give, in the absence of any
other, by saying that it is _the science of magnitudes_, or, what is
more positive, _the science which has as its goal the measurement of magnitudes_. This
scholastic insight has, undoubtedly, singularly need to acquire more
precision and more depth. But the idea is basically correct; it
is even sufficiently extensive, when it is properly conceived. It is
important, moreover, in such matters, when one can without
inconvenience, to rely on generally accepted notions.
So let's see how, starting from this rough outline, we can rise to
a true definition of mathematics, to a definition that is
worthy of corresponding to the importance, extent and difficulty of
science.
The question of _measuring_ a quantity does not in itself present to
the mind any other idea than that of the simple immediate comparison of
this quantity with another similar quantity supposedly known, which we
take for _unit_ between all those of the same. species. Thus, when
we limit ourselves to defining mathematics as having for its object the measurement
of magnitudes, we give a very imperfect idea of it, for it is even
impossible to see by this how there is place, in this respect, for
any science whatever. , and above all to a science as vast and as
deep as mathematical science rightly reputed to be.
instead of an immense chain of very prolonged rational works, which
provide our intellectual activity with inexhaustible nourishment,
science would seem only to consist, according to such a statement, in a
simple series of mechanical processes, to obtain directly, at the 'using
operations analogous to the superposition of lines, the ratios of the
quantities to be measured to those by which they are to be measured.
Nevertheless, this definition has no other fault than
that it is not sufficiently thorough. It does not mislead as to
the true final goal of mathematics; only it presents as
direct an object which, almost always, is, on the contrary, strong
indirect, and by that, it in no way makes us conceive of the nature of
science.
In order to achieve this, we must first consider a general fact, very easy
to ascertain. This is because the _direct_ measurement of a quantity, by
superposition or by some similar process, is most often for
us a completely impossible operation: so that if we had
no other means of determine the magnitudes as
immediate comparisons , we would be forced to give up the knowledge of
most of those that interest us.
We will understand all the accuracy of this general observation, by
limiting ourselves to considering especially the particular case which presents
obviously the easiest, that of measuring a straight line
by another straight line. This comparison, which, of all those
we can imagine, is arguably the simplest, can
almost never be done immediately. By reflecting
on the set of conditions necessary for a straight line to be
capable of direct measurement, we see that most often they
cannot be fulfilled at the same time, relatively to the lines that we
wish to know. The first and coarsest of these conditions,
that of being able to travel the line from end to end, in order to
successively carry the unit to its full extent, obviously already excludes the
most of the distances which interest us the most; first
all the distances between the different celestial bodies, or from the earth
to some other celestial body, and then even most of the
terrestrial distances , which are so frequently inaccessible. When this first
condition is fulfilled, the length must still be neither
too great nor too small, which would also make direct measurement
impossible; it must be suitably located, etc. The
slightest circumstance, which abstractly would not seem
likely to introduce any new difficulty, will often suffice, in reality, to
prevent us from any direct measurement. So, for example, such line that we
We could measure exactly with the greatest facility, if it were
horizontal, it will suffice to conceive it upright vertically, for
measurement to become impossible. In short, the immediate measurement
of a straight line presents such a complication of difficulty,
especially when we want to bring some precision to it, that
we hardly ever come across other lines capable of being measured
directly with precision, at least. among those of a certain
size, only purely artificial lines, created expressly
by us to include a direct determination, and to which we
manage to relate all the others.
What I have just established relatively to lines is conceived, with much
more reason, of surfaces, volumes, speeds, times,
forces, etc., and, in general, of all the other quantities
susceptible of an exact appreciation, and which, by their nature,
necessarily present many more obstacles still to immediate measurement.
It is therefore useless to stop there, and we must consider as
sufficiently established the impossibility of determining, by measuring them
directly, most of the quantities that we wish to know. It is
this general fact that requires the formation of mathematical science,
as we will see. Because, renouncing, in almost all cases,
the immediate measurement of magnitudes, the human mind must have sought to
determine them indirectly, and this is how it was led to the
creation of mathematics.
The general method which one constantly employs, the only one evidently which one
can conceive, to know magnitudes which do not involve
a direct measurement, consists in relating them to others which are
capable of being determined immediately, and of after which we
succeed in discovering the former, by means of the relations which exist
between them. This is the precise object of
mathematical science considered as a whole. To get an idea
sufficiently extensive, it must be considered that this
indirect determination of the magnitudes may be indirect to very different degrees.
In a great number of cases, which are often the most important, the
quantities, to the determination of which we reduce the search for the
principal quantities that we want to know, cannot themselves
be measured immediately, and must consequently, to in turn,
become the subject of a similar question, and so on; so
that, on many occasions, the human mind is obliged to establish a
long series of intermediaries between the system of unknown quantities
which are the final object of its research, and the system of
quantities susceptible of direct measurement, according to which
the former are finally determined, and which at first appear to have
no connection with them.
A few examples will suffice to clarify what the
preceding generalities could present that is too abstract.
Let us consider, in the first place, a very simple natural phenomenon which
may nevertheless give rise to a real mathematical question
capable of effective applications, the phenomenon of the
vertical fall of heavy bodies.
By observing this phenomenon, the mind most foreign to
mathematical conceptions immediately recognizes that the two quantities it
present, namely: the height from which a body has fallen, and the time of its
fall, are necessarily linked to each other, since they vary
together, and remain fixed simultaneously; or, according to the language of
geometers, that they are _function_ of one another. The phenomenon,
considered from this point of view, therefore gives rise to a
mathematical question , which consists in supplementing the direct measurement of one of
these two quantities when it is impossible, by the measurement of
the other. It is thus, for example, that we can indirectly determine
the depth of a precipice, by limiting ourselves to measuring the time that a
body would take in falling to the bottom; and, by proceeding
suitably, this inaccessible depth will be known with just as
much precision as if it were a horizontal line placed under the
circumstances most favorable to easy and exact measurement. On
other occasions, it is the height from which the body has fallen which will be
easy to know, while the time of the fall could not
be observed directly; then the same phenomenon will give rise to the
inverse question, to determine the time according to the height; as, for
example, if we wanted to know what would be the duration of the
vertical fall of a body falling from the moon to the earth.
In the previous example, the mathematical question is very simple, from
less when we do not have regard to the variation of intensity of gravity,
nor to the resistance of the fluid which the body passes through in its fall. But,
to enlarge the question, it will suffice to consider the same phenomenon
in its greatest generality, supposing the oblique fall, and taking into
account all the main circumstances. So, instead of
simply offering two variable quantities linked together by an
easy-to-follow relation , the phenomenon will present a greater number,
the space traveled, either vertically or
horizontally, time employed in traversing it, the speed of the body at
each point of its course, and even the intensity and direction of its
primitive impulse, which could also be considered as variables,
and finally, in certain cases, to take everything into account, the resistance of the
medium and the energy of gravity. All these various quantities will be
linked together, so that each in turn can be
determined indirectly from the others, which will present as many
distinct mathematical researches as there will be
coexisting magnitudes in the phenomenon considered. This very simple change in
the physical conditions of a problem may cause, as it happens in
fact for the example cited, that a mathematical research, originally
very elementary, suddenly ranks among the most common questions.
difficult, the complete and rigorous solution of which has so
far surpassed all the greatest forces of the human mind.
Let us take a second example in geometric phenomena. Whether it
is a question of determining a distance which is not susceptible of
direct measurement ; it will generally be conceived as part of a
_figure_, or of some system of lines, chosen in such a way
that all its other elements can be observed immediately; for
example, in the simplest case and to which all the others can be
reduced finally, we will consider the distance proposed as
belonging to a triangle, in which we could determine
directly, either another side and two angles, or two sides and a
single angle. Consequently, the knowledge of the sought distance, instead
of being obtained immediately, will be the result of a mathematical work
which will consist in deducing it from the elements observed, according to the relation
which links it with them. This work could become successively more
and more complicated, if the elements supposedly known could, in their
turn, as it happens most often, be determined only in an
indirect way , with the aid of new auxiliary systems, of which the number,
in large operations of this kind, ends by becoming sometimes
very considerable. The distance once determined, this only
knowledge will often suffice to obtain new
quantities, which will offer the subject of new mathematical questions.
Thus, when we know at what distance an object is located, the simple
observation, always possible, of its apparent diameter, must
obviously make it possible to determine indirectly, however inaccessible
it may be, its real dimensions, and, by a consequence of
analogous researches, its surface, its volume, its very weight, and a
host of other properties, the knowledge of which seemed necessarily to
be forbidden to us.
It is by such works that man has been able to come to know,
not only the distances of the stars from the earth, and consequently,
between them, but their effective size, their true figure, even the
inequalities of their surface, and, which seems to elude even more
from our means of investigation, their respective masses, their
average densities, the main circumstances of the fall of
heavy bodies to the surface of each of them, etc. By the power of
mathematical theories , all these various results, and many others
relating to the different classes of natural phenomena,
definitively required no other immediate measurements than those of a very small
number of straight lines, suitably chosen, and a greater
number of angles. We can even say, strictly speaking, to indicate a
only feature the general scope of science, that if one did
not fear with reason of multiplying without necessity the
mathematical operations , and if, consequently, one should not reserve them
only for the determination of the quantities which could
by no means be measured directly, or in a fairly exact manner, the
knowledge of all the quantities capable of precise estimation
that the various orders of phenomena can offer us, would
ultimately be reducible to the immediate measurement of a single straight line and
a number of suitable angles.
We have therefore now succeeded in precisely defining
mathematical science , by assigning it as its goal, the _indirect_ measurement of
sizes, and saying that we constantly propose to _determine the
sizes one by another, according to the precise relations that
exist between them. This statement, instead of just giving
the idea of an art, as all
ordinary definitions have done up to now , immediately characterizes a real science, and
shows it on the spot composed of an immense chain of operations.
intellectual, which could obviously become very complicated,
because of the series of intermediaries that will have to be established between the
unknown quantities and those which include a direct measurement, the
number of co-existing variables in the proposed question, and the
nature of the relations which
the phenomena considered will furnish between all these various magnitudes . According to such a definition, the
mathematical mind consists in always looking at all
the quantities which any phenomenon can present as interrelated , with a view to
deducing them from each other. Now, there is obviously no
phenomenon which cannot give rise to considerations of this kind;
whence results the naturally indefinite extension and even the rigorous
logical universality of mathematical science: we will seek further
on to circumscribe as exactly as possible its effective extension.
The preceding explanations clearly establish the
justification of the name used to designate the science we are
considering. This denomination, which has taken on such a
definite meaning today , simply signifies _science_ in general. One
such designation, strictly accurate for the Greeks, who had
no other real science, has been retained by modern as
indicating mathematics as the _science_ par excellence. And, in
fact, the definition to which we have just been led, if we set
aside the circumstance of the precision of the determinations, is nothing
else than the definition of any true science whatever, for
each has not - it is not necessarily intended to determine
phenomena by one another, according to the relations which exist
between them? All _science_ consists in the coordination of facts; if
the various observations were entirely isolated, there would be no
_science_. It can even be said generally that _science_ is
essentially intended to dispense, as far as the
various phenomena allow, from any direct observation, by making it possible to
deduce from the smallest possible number of immediate data, the
greatest possible number of results. Is this not, in fact, the
real use , either in speculation or in action, of the _laws_ which we
manage to discover between natural phenomena? Science
mathematics does, according to this, only push to the highest
possible degree , both in terms of quantity and
quality, on the subjects truly within its purview, the same kind of
research that continues, to varying degrees more or less inferior, each
real science, in its respective sphere.
It is therefore by the study of mathematics, and only by it, that
one can get a fair and thorough idea of what a
_science_ is. It is only there that one should seek to know with
precision the general method which the human mind constantly employs
in all its positive researches, because nowhere else
questions are not answered so thoroughly, and
deductions extended so far with rigorous severity. It is there
also that our understanding has given the greatest proofs of its
force, because the ideas which it considers there are of the highest degree
of abstraction possible in the positive order. Any
scientific education which does not begin with such a study, therefore
necessarily sins from its base.
We have so far considered mathematical science only as a
whole, without regard to its divisions. We must
now, in order to complete this general view and to form a correct
idea of the philosophical character of science, consider its division
fundamental. The secondary divisions will be examined in the
following lessons .
This main division can only be truly rational, and
derive from the very nature of the subject, so long as it presents itself
spontaneously, by making the exact analysis of a
complete mathematical question . Thus, after having determined above what is the
general object of mathematical work, let us now characterize with
precision the various principal orders of research of which they
constantly consist.
The complete solution of any mathematical question
necessarily breaks down into two parts, of an essentially distinct nature,
and whose relation is invariably determined. Indeed, we have
seen that all mathematical research has for object to determine
unknown quantities, according to the relations which exist between them and
known quantities. However, to this end, it is obviously necessary first of all to
obtain a precise knowledge of the existing relations between the
quantities considered. This first order of research constitutes
what I call the _concrete_ part of the solution. When it is
finished, the question changes in nature; it is reduced to a pure
question of numbers, consisting now simply in determining
unknown numbers, when we know what precise relations link them to
known numbers. It is in this second order of research that
consists what I call the _abstract_ part of the solution. Hence
the fundamental division of general mathematical science into
two great sciences, abstract mathematics and
concrete mathematics .
This analysis can be seen in any
complete mathematical question , no matter how simple or complicated.
To make it clearly understood, it will suffice to indicate a single example.
Taking up the phenomenon already cited of the vertical fall of a
heavy body , and considering the simplest case, we see that in order to succeed
in determining one by the other the height from which the body has fallen and the
duration of its fall, we must begin by discovering the exact relation of
these two quantities, or, according to the language of geometers, the_equation_
which exists between them. Before this first research is
finished, any attempt to numerically determine the value of
one of these two quantities by that of the other would obviously be
premature, because it would have no basis. It is not enough to
vaguely know that they are dependent on each other, which everyone
sees right away, but it is necessary to determine what this
dependence consists of ; which can be very difficult, and in fact constitutes, in
the present case, the incomparably higher part of the problem. The
true scientific spirit is so modern and still so rare,
that perhaps no one before Galileo had even noticed
the increase in speed that a body experiences in its fall, which
excludes the hypothesis, towards which our intelligence, always
involuntarily carried to suppose in each phenomenon the _functions_ the
simplest, without any other motive than its greater facility to
conceive them, would be naturally involved, the height proportional to
time. In short, this first work leads to the discovery of the law
of Galileo. When this concrete part is finished, the research
becomes of a completely different nature. Knowing that the spaces traversed by
the body in each successive second of its fall grow like the
series of odd numbers, it is then a purely numerical
and abstract question to deduce from it either the height according to time, or
time by height, which will consist in finding that, according to the
established law , the first of these two quantities is a known multiple of the
second power of the other, from which we must finally conclude the
value of one when that of l other will be given.
In this example, the concrete question is more difficult than the
abstract question. It would be the reverse, if we considered the same
phenomenon in its greater generality, as I considered it more
up for another reason. Depending on the case, it will sometimes be the first,
sometimes the second of these two parts which will constitute the main
difficulty of the total question; the mathematical law of the phenomenon
being able to be very simple, but difficult to obtain, and, on other
occasions, easy to discover, but very complicated: so that the
two great sections of mathematical science, when compared
en masse , must be regarded as exactly equivalent in
extent and difficulty, as well as in importance, as we shall
see later by considering each of them separately.
These two parts, essentially distinct, according to the explanation
previous, by the object which the mind proposes therein, are not less so
by the nature of the researches of which they are composed.
The first must bear the name of _concrete_, because it obviously depends
on the kind of phenomena considered, and must necessarily vary
when new phenomena are considered; while the second is
completely independent of the nature of the objects examined, and relates
only to the numerical relations which they present, which should
call it _abstract_. The same relations can exist in a
large number of different phenomena, which, despite their extreme
diversity, will be considered by the surveyor as offering a question
analytical, susceptible, by studying it in isolation, to be resolved
once and for all. Thus, for example, the same law which reigns between
space and time, when one examines the vertical fall of a body
in the vacuum, is found for other phenomena which do not offer any
analogy with the first nor between them: because it also expresses the
relation between the area of a spherical body and the length of its
diameter; it also determines the decrease in the intensity of
light or heat due to the distance from illuminated or
heated objects , etc. The abstract part, common to these various
mathematical questions , having been treated on the occasion of only one of them,
being will be found, by that very fact, for all the others; while the
concrete part will necessarily have to be taken up for each
separately, without the solution of some being able to provide, in
this respect, any direct help for that of the following ones. It is
impossible to establish true general methods which, by a
determined and invariable course, ensure, in all cases, the
discovery of the existing relations between quantities, relative to
any phenomena: this subject necessarily includes only
special methods. for this or that class of
geometric, or mechanical, or thermological phenomena, etc. We can, at
On the contrary, whatever source the quantities considered come from,
establish uniform methods for deducing them from one another,
assuming their exact relationships are known. The abstract part of
mathematics is therefore, by its nature, general; the concrete,
special part.
By presenting this comparison from a new point of view, we can
say that concrete mathematics has an
essentially experimental, physical, phenomenal philosophical character ; while that of
abstract mathematics is purely logical, rational. This is not
the place to discuss exactly the procedures employed by the
human mind to discover the mathematical laws of phenomena. But, either
that precise observation itself suggests the law, or, as it
happens more often, that it only confirms the law constructed
by reasoning according to the most common facts; it is always
certain that this law is only considered real insofar as it
shows itself to be in agreement with the results of direct experience. Thus, the
concrete part of any mathematical question is necessarily founded
on the consideration of the outside world, and can never, whatever
the part of the reasoning, be solved by a simple
series of intellectual combinations. The abstract part, on the
contrary, when it has first been very exactly separated, cannot
to consist only in a series of more or less
prolonged rational deductions . Because, if we have once found the equations of a phenomenon,
the determination of the one by the other of the quantities
considered in it, some difficulties moreover that it can often
offer, is solely a matter of reasoning . It is for
the intelligence to deduce, from these equations,
results which are obviously included in them, although in a way that is perhaps
very implicit, without there being any need to consult the world again.
outside, whose consideration, henceforth become foreign, must even
be carefully discarded in order to reduce the work to its
own real difficulty.
We see, by this general comparison, of which I must confine myself here to
indicating the principal features, how natural and profound is the
fundamental division established above in mathematical science.
To complete the general exposition of this division, all that remains is
to circumscribe, as exactly as we can
in this first overview, each of the two main sections of
mathematical science .
The _ concrete mathematics_ having for object to discover the _equations_
of the phenomena, would seem, _a priori_, to have to consist of as many
distinct sciences as there are really different categories for
us among natural phenomena. But it is far from being
able to discover mathematical laws in all
orders of phenomena; we will even see later that, in this
respect, the greater part will very probably always elude
our efforts. In reality, in the present state of the human mind, there
are directly only two great general categories of phenomena whose
equations are constantly known; it is first the
geometric phenomena , and then the mechanical phenomena. So the
concrete part of mathematics therefore consists of geometry and
rational mechanics.
This is sufficient, it is true, to give it a complete character
of logical universality, when we consider all phenomena from the
highest point of view of natural philosophy. Indeed, if
all the parts of the universe were conceived as immobile, there
would obviously only be geometrical phenomena to be observed, since
everything would be reduced to relations of form, size, and
location; then having regard to the movements which are executed there, there is
reason to consider further mechanical phenomena. By applying here,
after having sufficiently generalized it, a philosophical conception,
due to M. de Blainville, and already cited for another use in the 1st
lesson (page 32), we can therefore establish that, seen under the static relation,
the universe presents only geometric phenomena; and, under the
dynamic relation, only mechanical phenomena. Thus geometry and
mechanics constitute, by themselves, the two
fundamental natural sciences , in the sense that all natural effects can be
conceived as mere necessary results, or laws of extent,
or laws of motion. .
But, although this conception is always logically possible, the
difficulty is to specialize it with the necessary precision, and to
follow it exactly in each of the general cases offered to us by the study of
nature, that is to say, to reduce effectively each
principal question of natural philosophy, for such
determined order of phenomena , to the question of geometry or mechanics, to which one
could rationally suppose it reduced. This transformation, which
requires beforehand great progress in the study of each class of
phenomena, has so far been really carried out only for
astronomical phenomena, and for a part of those considered by
terrestrial physics properly so called. This is how astronomy,
acoustics, optics, etc., finally became applications
of mathematical science to certain orders of observations [3]. But,
these applications not being, by their nature, strictly
circumscribed, it would be assigning to science an indefinite and
entirely vague domain to confuse them with it, as is done in
the ordinary division, so vicious to so many others respects,
pure and applied mathematics. We will therefore persist in looking at
concrete mathematics as uniquely composed of geometry and
mechanics.
[Note 3: I must make here, in anticipation, a
summary mention of thermology, to which I will
devote a special lesson later. The
mathematical theory of the phenomena of heat took, by the
memorable works of its illustrious founder, such a
character that we can today conceive it, after
geometry and mechanics, as a veritable third
section distinct from concrete mathematics, since M.
Fourier established, in a manner entirely direct, the
thermological equations, instead of
hypothetically representing the questions as applications of
mechanics, as has been tried to do for
electrical phenomena, for example. This great
discovery, which, like all those which relate to the
method, is not yet properly appreciated, deserves
especially our attention; for, in addition to its
immediate importance for the truly rational and positive study of such a
universal and fundamental order of phenomena,
it tends to raise our philosophical hopes, as to
the future extension of the legitimate applications of
mathematical analysis , thus which I will explain in the second
volume of this course, examining the general character of
this new series of works. I would not have hesitated
now to treat thermology, thus conceived, as a
third main branch of concrete mathematics, if
I had not feared to diminish the
deviating too much from ordinary habits.]
As to _abstract mathematics_, the
general division of which I will examine in the next lesson, its nature is clearly and exactly
determined. It is made up of what is called the _calcul_, taking
this word in its greatest extension, which embraces from the
simplest numerical operations to the most sublime combinations of
transcendent analysis. The _calcul_ has for proper object to solve
all the questions of numbers. Its starting point is, constantly and
necessarily, the knowledge of precise relations, that is to say
of_equations_, between the various quantities that are considered
simultaneously, which is, on the contrary, the term of
concrete mathematics . However complicated or indirect
these relations may be , the final goal of the science of _calculus_ is
always to deduce the values of unknown quantities from those of
known quantities. This _science_, although more perfected
than any other, is, without doubt, really little advanced yet,
so that this goal is seldom attained in a completely
satisfactory manner . But that is none the less its true character. To
clearly conceive of the true nature of a science, it must always be
assumed to be perfect.
In order to summarize as philosophically as possible the considerations
exposed above on the fundamental division of mathematics, it is
important to note that it is only an application of the
general principle of classification which allowed us to establish, in the
previous lesson , the rational hierarchy of the different
positive sciences .
If we compare, in fact, on the one hand calculation, and on the other hand
geometry and mechanics, we verify, in relation to the ideas
considered in each of these two main sections of
mathematics, all the essential characteristics of our
encyclopedic method . Analytical ideas are obviously at the same time more
abstract, more general and simpler than geometric ideas.
or mechanical. Although the main conceptions of
mathematical analysis , considered historically, were formed under
the influence of considerations of geometry or mechanics, to the
improvement of which the progress of calculus is closely linked,
analysis is nonetheless, under the logical point of view,
essentially independent of geometry and mechanics, while the
latter are, on the contrary, necessarily founded on the
former.
Mathematical analysis is therefore, according to the principles which we have
consistently followed hitherto, the true rational basis of the
entire system of our positive knowledge. It constitutes the first and the
most perfect of all the basic sciences. The ideas she
deals with are the most universal, the most abstract and the
simplest that we can actually conceive of. One could not attempt
to go further, in these three equivalent reports, without
inevitably falling into metaphysical reveries. For what
effective _substractum_ could remain in the mind to serve as a positive subject
for reasoning, if we wished to remove some further circumstance
in the notions of indeterminate, constant or variable quantities,
such as geometers use them today, in order to to rise to
a so-called higher degree of abstraction, as
ontologists believe ?
This peculiar nature of mathematical analysis makes it
easy to explain why, when properly employed, it
offers us such a powerful means, not only to give more precision
to our real knowledge, which is self-evident. - even, but
above all to establish an infinitely more perfect coordination in
the study of the phenomena which involve this application. For, the
conceptions having been generalized and simplified as much as possible, to
such a point that a single analytical question, resolved in the abstract,
contains the implicit solution of a host of
diverse physical questions , it must necessarily result for the human mind
greater ease in perceiving relations between phenomena which
at first seemed entirely isolated from one another, and from which we have
thus succeeded in extracting, in order to consider it separately, all that they have
in common. It is thus, in examining the progress of our intelligence
in the solution of important questions of geometry and mechanics,
we see naturally arise, through the intermediary of analysis,
the most frequent and the most unexpected comparisons between
problems which initially offered no apparent connection, and
which we often end up considering as identical.
Could we, for example, without the aid of analysis, perceive the
least analogy between the determination of the direction of a curve at
each of its points, and that of the speed acquired by a body at
each moment of its varied motion, questions which, however diverse
they may be, only make one, in the eyes of the surveyor?
The high relative perfection of mathematical analysis, compared with
all the other branches of our positive knowledge, is conceived
with the same facility, when we have fully grasped its true general character.
This perfection does not depend, as the metaphysicians, and
especially Condillac, after a superficial examination, have believed it , in the nature of the
eminently concise and general signs which one employs as instruments of
reasoning. On this important special occasion, as on all
others, the influence of signs has been greatly exaggerated, although
it is undoubtedly very real, as had been recognized, before
Condillac, and in a much more exact, most surveyors.
In reality, all the great analytical conceptions were formed
without the algebraic signs being of any essential help,
other than to exploit them after the mind had obtained them.
The superior perfection of the science of calculus is due mainly to
the extreme simplicity of the ideas which it considers, by some signs that
they are expressed: so that it does not
with the help of any artifice of scientific language, even
supposing it possible, to perfect, to the same degree, theories which,
bearing on more complex notions, are necessarily condemned,
by their nature, to a more or less logical inferiority smaller according to
the corresponding class of phenomena.
The examination which we have attempted to make, in this lesson, of the
philosophical character of mathematical science, would remain incomplete if, after
having considered it in its object and in its composition, we
did not indicate some general considerations directly relative
to the real extent of its domain.
To this end, it is essential to recognize above all, in order to
to get a correct idea of the true nature of mathematics, that,
from a purely logical point of view, this science is, by itself,
necessarily and rigorously universal. For there is no
question whatever which cannot ultimately be conceived as
consisting in determining quantities one by the other according to
certain relations, and, consequently, as reducible, in the final
analysis, to a simple question of numbers. We will understand this if we
actually notice that, in all our research, to whatever order
of phenomena they relate to, we definitely aim
to arrive at numbers, at doses. Although we do not succeed
more often than in a very crude manner and according to
very uncertain methods , it is none the less evident that such is the
real term of all our problems whatever. Thus, to take an example
from the class of phenomenon least accessible to the mathematical mind,
the phenomena of living bodies, considered even, for more
complication, in the pathological case, is it not obvious that
all questions therapy can be considered as
consisting in determining the quantities of all the various modifiers
of the organism which must act on it to bring it back to the normal state,
admitting, according to the use of geometers, the zero values,
negative, or even contradictory, for some of these quantities
in certain cases? Undoubtedly, such a way of representing the
question cannot in fact be really followed, as we will
see it, for the most complex phenomena, because it
presents us in the application of insurmountable difficulties; but when
it is a question of conceiving in the abstract all the intellectual scope
of a science, it is important to suppose it to be the total extension of which it
is logically susceptible.
One would object in vain to such a conception the
general division of human ideas according to the two categories of Kant, of
quantity, and quality, the first of which alone would constitute the
exclusive domain of mathematical science. The very development of
this science has long shown positively the scarcity of reality
of this superficial metaphysical distinction. For the
fundamental conception of Descartes on the relation of the concrete to the abstract in
mathematics, proved that all ideas of quality were
reducible to ideas of quantity. This conception, first established
by its immortal author, for geometrical phenomena only,
was then effectively extended by his successors to
mechanical phenomena ; and it comes to be nowadays to phenomena
thermological. As a result of this gradual generalization, there are
now no surveyors who do not consider it, in a purely
theoretical sense , to be applicable to all our real ideas
whatever, so that any phenomenon is logically capable
of being represented by an _equation_, as well as a curve or
movement, except the difficulty of finding it, and that of _resolving_,
which can be and often are superior to the greater forces of
the human mind.
But if, in order to form a suitable idea of mathematical science,
it is important to conceive it as necessarily endowed by its
nature with a rigorous logical universality, it does not
it is essential now to consider the great real limitations
which, in view of the weakness of our intelligence, singularly narrow
its effective domain, as the phenomena become more complicated by
specializing.
Any question can doubtless, as we have just seen, be
conceived as reducible to a pure question of numbers. But the
difficulty of really treating it from this point of view, that is to say
of effecting such a transformation, is all the greater, in the
various essential parts of natural philosophy, as we
consider more complicated phenomena, so that except for the
simplest and most general phenomena, it soon becomes
insurmountable.
We will easily feel this, if we consider that, in order to fit a
question into the domain of mathematical analysis, we must first
have succeeded in discovering precise relations between the
coexisting quantities in the phenomenon studied, the establishment of these equations
of phenomena being the necessary starting point of all
analytical work . Now, this must obviously be all the more difficult, as
it is a question of more particular phenomena, and consequently more
complicated. By examining from this point of view the various
fundamental categories of natural phenomena established in the preceding lesson,
we will find that, all things considered, it is
three first, comprising all the _inorganic physics_, that one can
legitimately hope to one day reach this high degree of
scientific perfection , at least as far as such a limit can be set with
precision. As I will later treat this discussion specially
in relation to each basic science, it will suffice to indicate it here
in the most general way.
The first condition for phenomena to include
mathematical laws capable of being discovered is obviously that the
various quantities which they present can give rise to
fixed numbers . However, by comparing, in this respect, the two main sections
principles of natural philosophy, we see that the whole _organic physics_
, and probably also the most
complicated parts of inorganic physics, are necessarily
inaccessible, by their nature, to our mathematical analysis, by virtue
of the extreme numerical variability of corresponding phenomena. Any
precise idea of fixed numbers is truly out of place in the
phenomena of living bodies, when one wishes to employ it other than
as a means of relieving attention, and attaching some importance
to the exact relations of the assigned values. In this respect, the
reflections of Bichat, on the abuse of the mathematical mind in
physiology, are perfectly correct; we know to what aberrations
this vicious way of considering living bodies has led.
The different properties of gross bodies, especially the most general,
present themselves in each of them with almost invariable degrees, or
at least they undergo only simple variations, separated by
long intervals of uniformity, and which it is possible, therefore,
to subject to precise and regular laws. Thus, the
physical qualities of an inorganic body, principally when it is solid, its
shape, its consistency, its specific gravity, its elasticity, etc.,
present, for a considerable time, a numerical fixity.
remarkable, which allows them to be considered really and usefully from
a mathematical point of view. We know that this is already very
much the case with the chemical phenomena presented by the same
bodies, and which, more complicated, depending on a much larger number of
circumstances, present more extensive and more frequent variations. ,
and consequently more irregular. Also, according to some considerations
already indicated in the first lesson (page 45) and which will be
specially developed in the third volume of this course, we
cannot only assure today, in a general way, that there
takes place in the conception of fixed numbers in chemistry, even in relation to
the simplest, as to the relative proportions of bodies in their
combinations, which clearly shows how far such an order of
phenomena is still from comprising true
mathematical laws . Let us nevertheless admit, for this case, the possibility and
even the future probability, in order not to make too detailed the
discussion of the general limit which it is a question of establishing here in relation
to the extension, actually possible, of the real field of
mathematical analysis . There will be no longer the slightest doubt as soon as we
pass to the phenomena presented by bodies, considered in this
state of continual internal agitation of their molecules, which
essentially constitutes what we call _life_, considered in
the most general way, in the set of beings who
manifest it to us . Indeed, a character eminently peculiar to
physiological phenomena , and which their more exact study now makes more
perceptible from day to day, is the extreme numerical instability which they
present, under whatever aspect one examines them, and which we will see
later, when the natural order of matters leads us there, to be a
necessary consequence of the very definition of living bodies. As for
now, it suffices to note this incontestable observation, verified
by all the facts, that every property whatever of an organized body,
whether geometric, mechanical, chemical, or vital, is
subject, in its quantity, to immense and
quite irregular numerical variations , which succeed each other at the closest intervals
under the influence of a host of circumstances , both
external and internal, variables themselves; so that any
idea of fixed numbers, and, consequently, of mathematical laws which we
may hope to obtain, really implies contradiction with the
special nature of this class of phenomena. Thus, when one wants to
evaluate with precision, even only the simplest qualities
of a living being, for example its average density, or that of one of
its principal constituent parts, its temperature, the speed of its
internal circulation, the proportion of the immediate elements which
compose its solids or its fluids, the quantity of oxygen which it
consumes in a given time, the mass of its absorptions or of its
continual exhalations, etc., and, a fortiori, the energy of
his muscular forces, the intensity of his impressions, etc., it is
not only necessary , which is evident, to do, for each of these results,
as many observations as there are species or races and varieties
in each species; we must also measure the
very considerable change which this quantity undergoes in passing from an individual to
another, and, as to the same individual, according to his age, his state of health
or illness, his interior disposition, the
incessantly mobile circumstances of all kinds under the influence of which he is
placed, such as the atmospheric constitution, etc. . What then can
mean these so-called numerical evaluations so carefully
recorded for the various physiological or even
pathological phenomena , and deduced, in the most favorable case, from a single
real measurement, when a multitude of them are needed? They can only
mislead on the true course of the phenomena, and must
be applied rationally only as a means, so to speak.
mnemonic, to fix ideas. In any case, there is obviously a
total impossibility of ever obtaining true mathematical laws.
It is still more strongly the same for social phenomena, which
offer an even greater complication, and, consequently, a
greater variability, as we will establish especially in the
fourth volume of this course.
It is not, however, that we should cease, according to this, to conceive,
as a general philosophical thesis, the phenomena of all orders as
necessarily subject by themselves to mathematical laws, which we
are only condemned to ignore. still in most cases, at
because of the too great complication of the phenomena. There is in fact
no reason to think that, in this respect, the most
complex phenomena of living bodies are essentially of a
special nature other than the simplest phenomena of gross bodies. For, if it
were possible to rigorously isolate each of the simple causes which
combine to produce the same physiological phenomenon, everything
suggests that it would show itself endowed, in determined circumstances,
with a kind of influence and a quantity of action as exactly fixed
as we see it in universal gravitation, a true type of
the fundamental laws of nature. What generates variability
irregular effects, it is the great number of different agents determining
at the same time the same phenomenon, and from which it results that, in
very-complicated phenomena , there are perhaps not two strictly
similar cases. . To find such a difficulty, we do not need
to go as far as the phenomena of living bodies. It already presents itself
in those of gross bodies, when we consider the most
complex cases; for example, by studying weather phenomena. There
can be no doubt that each of the many agents which contribute to the
production of these phenomena is separately subject to
mathematical laws , although we still do not know most of them;
but their multiplicity makes the observed effects as irregularly
variable as if each cause were not subject to any
precise condition .
The preceding consideration leads to perceive a second
distinct reason by virtue of which it is necessarily forbidden for us, given the
weakness of our intelligence, to bring the study of
the most complicated phenomena into the field of the applications of
mathematical analysis . Indeed, regardless of what, in the
most special phenomena , the actual results are so variable that
we cannot even enter fixed values in them, it follows from the
complication of the cases, which, even so we could one day know
the mathematical law to which each agent taken separately is subjected, the
combination of such a large number of conditions would make the
corresponding mathematical problem so superior to our weak means that
the question would most often remain insoluble. It is not thus
that one can make a real and fruitful study of the major part of
natural phenomena.
In order to appreciate this difficulty as exactly as possible, let us
consider how complicated mathematical questions become,
even relative to the simplest phenomena of gross bodies, when
we want to bring the abstract state sufficiently close to the concrete state, in
having regard to all the principal conditions which may exercise on
the effect produced, a real influence. We know, for example, that the
very simple phenomenon of the flow of a fluid, by virtue of its
gravity alone , through a given orifice, has so far not had a
complete mathematical solution , when we want to hold account of all the
essential circumstances. It is still so, even for the
still simpler movement of a solid projectile in a
resistant medium .
Why has mathematical analysis been able to adapt, with
such admirable success , to the in-depth study of celestial phenomena? Because
they are, despite their vulgar appearances,
all the others. The most complicated problem they present, that
of the modification produced, in the movement of two bodies tending
towards each other by virtue of their gravitation, the influence of a
third body acting on both of them. the same way, is much
less composed than the simplest earthly problem. And, nevertheless,
it already presents such a difficulty that we still have only
approximate solutions. It is even easy to see, by examining this
subject more deeply, that the high perfection to which
solar astronomy has been able to rise by the use of mathematical science
is still essentially due to what we have skillfully taken advantage of.
of all the special and, so to speak,
accidental facilities which the
special constitution , very favorable in this respect, of our planetary system offered for the solution of problems .
In fact, the planets of which it is composed are relatively few in number,
but above all they are, in general, of very unequal masses and much
less than that of the sun, and more distant from each
other; they have almost spherical shapes; their orbits are
almost circular, and have slight mutual inclinations,
etc. It follows from this set of circumstances that the disturbances
are most often not very considerable, and that to calculate them it is
It is usually sufficient to take into account, concurrently with the action of the
sun on each planet in particular, the influence of a single
other planet, capable, by its size and its proximity, of
determining sensible disturbances. But if, instead of such a state of
things, our solar system had been composed of a greater number of
planets concentrated in a lesser space, and nearly equal in
mass; if their orbits had offered very
different inclinations , and considerable eccentricities; if these bodies had
been of a more complicated form, for example,
very eccentric ellipsoids , etc .; it is certain that assuming the same law
real gravitation, we would not yet have succeeded in subjecting
the study of celestial phenomena to our mathematical analysis, and
probably we would not even have been able to untangle the
main law until now .
These hypothetical conditions would be precisely realized to
the highest degree in chemical phenomena, if we wanted to
calculate them according to the theory of general gravitation.
By properly weighing the various considerations which precede, one
will be convinced, I believe, that by reducing to the various parts of
inorganic physics the future extension of the great
really possible applications of mathematical analysis, I have rather exaggerated
that narrowed the extent of its effective domain. As much as it was important to
make sensitive the rigorous logical universality of
mathematical science , so much I had to point out the conditions which limit for
us its real extension, in order not to contribute to exclude the
human mind from the true scientific direction in the study of
most complicated phenomena, by the chimerical search for an
impossible perfection.
Thus, while endeavoring to enlarge as much as possible the
real field of mathematics, one must recognize that the most
difficult sciences are destined, by their nature, to remain indefinitely in
that preliminary state which prepares for the others. the
become accessible to mathematical theories. We must, for the
most complicated phenomena, be satisfied with analyzing with
exactitude the circumstances of their production, to relate
them to each other in a general way, to know the kind of influence
exerted by each principal agent, etc .; but without studying them from the
point of view of quantity, and consequently without hope
of introducing, into the corresponding sciences, that high degree of
perfection which, as to the simplest phenomena, procures a
suitable use of mathematics, namely in terms of the precision of
our knowledge, that is, what is perhaps even more remarkable,
with regard to their coordination.
It is through mathematics that positive philosophy began to be
formed: it is from them that the _method_ comes to us. It was therefore
naturally inevitable that, when the same way of proceeding had to be
extended to each of the other fundamental sciences, an effort should be made to
introduce the mathematical spirit into it to a greater degree than
the corresponding phenomena included; which then gave rise to
more or less extensive purification works, like those of Berthollet
on chemistry, to free themselves from this exaggerated influence. But each
science, while developing, has subjected the general positive method to
modifications determined by the phenomena which are peculiar to him,
from which results his special genius; it was only then that it took on its
true final character, which should never be confused with
that of any other basic science.
Having exposed, in this lesson, the essential aim and the
principal composition of mathematical science, as well as its general relations
with the whole of natural philosophy, its philosophical character
is determined, as far as it can be by a such preview. We
must now proceed to the special examination of each of the three great
sciences of which it is composed, calculus, geometry and
mechanics.
FOURTH LESSON.
SUMMARY. General view of Mathematical Analysis.
In the historical development of mathematical science since
Descartes, the progress of the abstract part has almost always been
determined by that of the concrete part. But it is none the less
necessary, in order to conceive of science in a truly
rational manner , to consider calculus in all its
main branches before proceeding to the philosophical study of geometry and
mechanics. Analytical theories, simpler and more
general than those of concrete mathematics, are by
themselves essentially independent of it; while these have, at
Contrary to their nature, a continual need for the former, without
whose help they could make almost no progress.
Although the principal conceptions of analysis still preserve
today some very sensible traces of their geometrical origin
ou mécanique, elles sont maintenant néanmoins essentiellement dégagées
of this primitive character, which is hardly manifested except for a
few secondary points; so that,
especially since the works of Lagrange, it is possible, in a dogmatic exhibition, to
present them in a purely abstract manner, in a single and
continuous system. This is what I will undertake in this lesson and the
next five, limiting myself, as befits the nature of this lesson.
course, to the most general considerations on each
major branch of computational science.
The final goal of our research in concrete mathematics being
the discovery of _equations_, which express the mathematical laws of the
phenomena considered, and these _equations_ constituting the real
starting point of the calculation, the object of which is to deduce the determination of the
quantities one by the other, I believe it is essential, before going
further, to deepen, more than is customary, this
fundamental idea of_equation_, a continual subject, either as a term or as an
origin , of all mathematical work. Besides the advantage of better
circumscribing the real field of analysis, this will
necessarily result in this important consequence of tracing
more exactly the real line of demarcation between the concrete part and
the abstract part of mathematics, which will complete the
general exposition of the fundamental division established in the previous lesson.
We usually form a much too vague idea of what
an _equation_ is, when we give this name to any kind of relation
of equality between two functions _any_ of the magnitudes that we are
considering. For, if any equation is obviously a relation of equality,
it is far from being, conversely, any relation of equality
or a real _equation_, of the kind to which, by their
nature, analytical methods are applicable.
This lack of precision in the logical consideration of a notion so
fundamental in mathematics, entails the serious drawback of rendering
almost inexplicable, in general thesis, the immense and
capital difficulty which we experience in establishing the relation of the concrete to
the abstract. , and which is commonly brought out with so much reason for
each great mathematical question taken separately. If the meaning of the word
_equation_ were really as extensive as is usually supposed
when defining it, we do not see, in fact, how great
difficulty might actually be, in general, establishing the
equations of some problem. For everything would thus seem to consist
of a simple question of form, which should never even require
great intellectual effort, since we can hardly
conceive of a precise relation which is not immediately a certain
relation of equality, or which does not can be promptly brought back to it by
some very easy transformations.
Thus, by admitting, in general, in the definition of _equations_,
all kinds of _functions_, we do not in any way account for the extreme
difficulty that we most often experience in putting a problem into
equation, and which is so frequently comparable to the effort involved in
the analytical development of the equation once obtained. In a word,
the abstract and general idea given of the_equation_ does not correspond
at all to the real meaning that geometers attach to this expression
in the effective development of science. There is here a logical
flaw , a lack of correlation, which it is very important to rectify.
To achieve this, I first distinguish two kinds of _functions_:
_abstract_, analytical functions, and _concretes_ functions. The
former can only enter the real _équations_ in
so we can now define, in an accurate and
sufficiently thorough, any _equation_: a relation of equality between
two _abstract_ functions of the quantities considered. In order not to have
to come back to this fundamental definition, I must add here,
as an essential complement without which the idea would not be
general enough, that these abstract functions can relate
not only to the quantities that the problem presents in effect of
itself, but also to all the other auxiliary quantities which are
attached to it, and which one can often introduce, simply by
mathematical artifice , in the sole view of facilitating the discovery of the equations
of the phenomena. I do here, in this explanation, only borrow
summarily, in anticipation, the result of a very important general discussion
, which will be found at the end of this lesson.
Let us now return to the essential distinction of functions in
abstract and concrete.
This distinction can be established by two essentially
different routes , complementary to each other; _a priori_, and _a
posteriori_: that is to say, by characterizing in a general way the
proper nature of each species of functions, and then by doing, what
is possible, the effective enumeration of all the
abstract functions known today, at least as regards the elements of which they
are composed.
_A priori_, the functions which I call _abstract_ are those which
express between magnitudes a mode of dependence that one can conceive
only between numbers, without it being necessary to indicate any
phenomenon whatever where it is realized. On the contrary, I name
_concrete_ functions those for which the mode of dependence
expressed cannot be defined or conceived except by assigning a
determined physical , geometrical, mechanical, or any other nature, in which
it actually takes place.
Most of the functions, at their origin, even those which are
today the most purely _abstract_, began by being
_concrete_; so that
above, limiting themselves to citing the various successive points of view
from which, as science has been formed, geometers have
considered the simplest analytical functions. I will give as an
example the powers, which have generally become abstract functions,
only since the work of Viète and Descartes. These functions x ^ 2, x ^ 3,
which in our present analysis are so well conceived of as simply
_abstract_, were, for the geometers of antiquity, only
entirely _concrete_ functions, expressing the relation of the
area d 'a square or the volume of a cube the length of their side.
They had such a character so exclusively in their eyes that it is
only according to their geometrical definition that they had discovered
the elementary algebraic properties of these functions, relative to
the decomposition of the variable into two parts, properties which
were, at that time, only true theorems of geometry, to which
we did not attach a digital sense until much later.
I shall have occasion to cite later, for another reason, a
new example very apt to make clearly felt the
fundamental distinction which I have just explained; it is that of
circular functions , either direct or inverse, which are still today
sometimes concrete, sometimes abstract, depending on the point of view from which
they are considered.
Considering now, _a posteriori_, this division of functions,
after having established the general character which makes a function abstract
or concrete, the question of knowing whether a given function is
truly abstract, and therefore capable of entering into real
analytical equations , will become a simple question of fact, since
we are going to enumerate all the functions of this kind.
At first glance, this enumeration seems impossible, the
distinct analytical functions being obviously in infinite number. But, by
dividing them into _simples_ and _composées_, the difficulty disappears. Because, if
the number of the various functions considered in the mathematical analysis
is really infinite, they are, on the contrary, even today,
composed of a very small number of elementary functions, which can
easily be assigned, and which are obviously sufficient to decide on the
abstract or concrete character of a given function, which will be of one or
the other nature, depending on whether it is composed exclusively of these
simple abstract functions, or whether it includes others. Here is
the table of these fundamental elements of all our
analytical combinations , in the present state of science. Obviously
, only the functions of a single variable should be considered for this purpose; those
relating to several independent variables being constantly, for
their nature, more or less _composées_.
Let x be the independent variable, y the corelative variable which depends on it.
The different simple modes of abstract dependence that we can
now conceive between y and x, are expressed by the following ten
elementary formulas , in which each function is coupled
with its _inverse_, that is to say, with that which would take place, according to
the _direct_ function, if we related x to y, instead of relating y
to x:
1st pair / left / begin {array} {ll} 1 ^ {/ rm o} /; y = a + x & / ldots /
/ mbox {function} sum, // 2 ^ {/ rm o} /; y = ax & / ldots / / mbox {function
/ em {diff / 'erence}}, / end {array} / right.
2nd couple / left / begin {array} {ll} 1 ^ {/ rm o} /; y = ax & / ldots /
/ mbox {function} product, // 2 ^ {/ rm o} /; y = / frac {a} {x} & / ldots /
/ mbox {function} quotient, / end {array} / right.
3rd couple / left / begin {array} {ll} 1 ^ {/ rm o} /; y = x ^ a & / ldots /
/ mbox {function} power, // 2 ^ {/ rm o} /; y = / sqrt [a] {x} & / ldots /
/ mbox {function} root, / end {array} / right.
4th couple / left / begin {array} {ll} 1 ^ {/ rm o} /; y = a ^ x & / ldots /
/ mbox {function} exponential, // 2 ^ {/ rm o} /; l_ax & / ldots /
/ mbox {function} logarithmic, / end {array} / right.
5th pair [4] / left / begin {array} {ll} 1 ^ {/ rm o} /; y = / mbox {sin} x & / ldots /
/ mbox {/ rm function} circular /; direct, // 2 ^ {/ rm
o} /; y = / mbox {arc (sin} = x) & / ldots / /
mbox inconnu funcionbcirculaire /; inverse./end Danemarkarray rire / right.
[Note 4: In the view of
The insufficient resources and extent of
mathematical analysis , the geometers count this last pair of
functions among the analytical elements. Although this
registration is strictly legitimate, it is important to
notice that the circular functions are not
exactly in the same case as the other
elementary abstract functions . There is between them this
very essential difference , that the functions of the first four
couples are really at the same time simple and abstract,
while the circular functions, which can manifest
successively one and the other character according to the point of
The view under which they are considered and the manner in which they
are employed, never present these two
properties simultaneously .
The sin x function is introduced into the analysis as a
new simple function, when it is conceived only
as indicating the geometric relation from which it derives;
but then it is obviously only a _concrete_ function.
In other circumstances, it analytically fulfills the
conditions of a true _abstract_ function, when
we consider sin x only as the shortened expression of the
formula / frac {e ^ {x / sqrt {-1}} - e ^ {- x / sqrt {-1}}} {2 / sqrt {-1}} or
of the equivalent series; but under this last point of view,
it is no longer really a new analytical function,
since it is presented only as a compound of the
preceding ones.
Nevertheless, circular functions have some
special qualities which enable them to be kept in the table of
rational elements of mathematical analysis.
1 ° They are subject to evaluation, although they retain
their concrete character; which allows them to be introduced
into the equations, as long as they relate only to
data, without it being necessary to have regard to their
algebraic expression.
2 ° We know how to carry out on the different
circular functions , compared with each other only, a certain
series of transformations, which do not require
knowledge of their analytical definition. This
obviously results in the faculty of introducing these functions into the
equations, even with respect to the unknowns, provided that there are
not concurrently non-trigonometric functions
of the same variables.
It is therefore only in cases where the
circular functions , relative to the unknowns, are combined in
the equations with abstract functions of another
species, that it is essential to have regard to their
algebraic interpretation in order to be able to solve the
equations, and consequently they cease to be
treated as new simple functions. But
even then , provided that this interpretation is taken into account,
their admission does not prevent the relations from having the
character of true analytical _equations_, which is
here the essential aim of our enumeration of
elementary abstract functions .
It should be noted, from the considerations indicated
in this note, that several other concrete functions
can usefully be introduced among the
analytical elements , if the principal conditions posed above
for the circular functions have been previously well
fulfilled. It is thus, for example, that the works of M.
Legendre, and recently those of M. Jacobi, on
_elliptical_ functions , have really enlarged the field of analysis;
it is the same for some definite integrals obtained
by M. Fourier, in the theory of heat.]
Such are the very few elements which directly compose all
the abstract functions known today. Somewhat multiplied
whatever they are, they are evidently sufficient to give rise to a
completely infinite number of analytical combinations.
No rational consideration strictly circumscribes _a
priori_ the preceding table, which is only the effective expression of
the current state of science. Our analytical elements are today
more numerous than they were for Descartes, and even for Newton and
Leïbnitz; It is at most a century since the last two couples
were introduced into the analysis by the works of Jean Bernouilli and
Euler. No doubt new ones will be admitted later; but,
as I will indicate at the end of this lesson, we cannot
hope that they will never be greatly multiplied, their real increase.
donnant lieu à de très-grandes difficultés.
We can therefore now form a positive idea, and nevertheless
sufficiently extensive, of what geometers understand by a
true _equation_. This explanation is eminently suited to
making us understand how difficult it must be to really establish the
_equations_ of phenomena, since we have only actually succeeded
when we have been able to conceive the mathematical laws of these phenomena with
the aid of functions entirely composed of the only analytical elements
which I have just enumerated. It is clear, in fact, that it is only
then that the problem really becomes _abstract_, and is reduced to a
pure question of numbers, these functions being the only
simple relations that we know how to conceive between the numbers, considered in
themselves. Until this time of the solution, whatever
appearances, the question is still essentially concrete, and does
not come within the domain of _calcul_. Now, the fundamental difficulty
of this passage from the _concret_ to the_abstract_ consists above all, in general,
in the insufficiency of this very small number of analytical elements which
we possess, and according to which nevertheless, in spite of the little variety
real that they offer us, it is necessary to succeed in representing all
the precise relations that can manifest to us all the differences
natural phenomena. Considering the infinite diversity which must necessarily
exist in this respect in the external world, it is
easy to understand
how our conceptions must frequently find themselves below the real difficulty; especially if we add that, these elements of our
analysis having been originally furnished to us by the
mathematical consideration of the simplest phenomena, since they all have,
directly or indirectly, a geometric origin, we have _a
priori_ no guarantee rational of their aptitude to
represent the mathematical laws of any other class of phenomena.
I will soon expose the general artifice, so deeply
ingenious, by which the human mind has succeeded in
singularly reducing this fundamental difficulty presented by the relation of the
concrete to the abstract in mathematics, without, however, having been
necessary to multiply the number of these analytical elements.
The preceding explanations precisely determine the real
object and the real field of abstract mathematics; I must
now turn to the consideration of its main divisions, for we have
so far always considered the _calcul_ as a whole.
The first direct consideration to be presented on the composition of the
science of _calcul_, consists in dividing it first into two branches
principal, to which, for lack of more suitable denominations, I will
give the names of _algebraic calculation_ or _algebra_, and of _
arithmetic_ or _arithmetic_ calculation, but by warning to take these two
expressions in their most extended logical sense, instead of the
much too restricted meaning that we usually attach to them.
The complete solution of any _calculus_ question, from the most
elementary to the most transcendent, necessarily consists of
two successive parts, the nature of which is essentially distinct.
In the first, we aim to transform the
proposed equations , so as to highlight the mode of formation of
quantities unknown by known quantities; this is what constitutes the
_algebraic_ question. In the second, we have in order to_evaluate_ the
_formulas_ thus obtained, that is to say, to immediately determine the
value of the numbers sought, already represented by certain
explicit functions of the given numbers; such is the _arithmetic_ question [5].
We see that, in any truly rational solution, it
necessarily follows the algebraic question, of which it forms the
indispensable complement , since it is obviously necessary to know the generation of the
numbers sought before determining their effective values for
each particular case. Thus, the term of the algebraic part becomes
the starting point of the arithmetic part.
[Note 5: Suppose, for example, that a question
provides between an unknown quantity x and two
known quantities a and b the equation: / [x ^ 3 + 3ax = 2b /] as it
would happen for the trisection of a angle. We see, immediately,
that the dependence between x on the one hand, and a, b on the other,
is completely determined; but, as long as the equation
retains its original form, we do not see in
what way the unknown derives from the data.
However, this is what must be discovered before thinking of
evaluating it. This is the object of the algebraic part of the
solution. When, by a series of transformations which have
successively made this derivation more and more
sensitive, we have come to present the proposed equation in
the form / x = / sqrt [3] {b + / sqrt {b ^ 2 + a ^ 3}} +
/ sqrt (3) {b- / sqrt {b ^ 2 + a ^ 3}} / the role of algebra is
ended; and even if we could not perform the
arithmetic operations indicated by this formula, we
would still have obtained a very real and
often very important knowledge of it. The role of arithmetic
will now consist, starting from this formula, to
find the number x when the values of the numbers a and b
will have been fixed.]
The _algebraic_ calculation and the _arithmetic_ calculation thus differ
essentially by the goal which one proposes there. They do not differ less
in the point of view from which we consider the quantities,
considered in the first as regards their relations, and in the
second as regards their values. The true spirit of _calculus_, in
general, demands that this distinction be maintained with the most severe
exactitude, and that the line of demarcation between the two eras of the
solution be made as sharp as the
proposed question permits . The careful observation of this precept, too little understood, can
be of useful help in each particular question, by directing
the efforts of our mind, at any time during the solution,
towards the corresponding real difficulty. In truth, the imperfection
of the science of calculus often obliges, as I will explain in the
following lesson, to very frequently mix algebraic
and arithmetical considerations for the solution of the same
question. But, although it is then impossible to divide the whole
of the work into two clearly defined parts, one purely
algebraic, and the other purely arithmetic, one can always avoid,
using the preceding indications, to confuse the two orders of
considerations, however intimate their mixture may ever be.
By trying to summarize as succinctly as possible the distinction that
I have just established, we see that_algebra_ can be defined, in general,
as having for object the _resolution_ of _equations_, which, although
initially appearing too restricted, is nonetheless sufficiently extended,
provided that these expressions are taken in all their logical sense,
which means transforming _implicit_ functions
into equivalent _explicit_ functions : likewise, the_arithmetic_ can be defined
as intended for the_evaluation_ of functions. Thus, by contracting the
expressions to the highest degree, I believe I can give a clear
idea of this division, by saying, as I will do henceforth
to avoid explanatory paraphrases, that the_algebra_ is the _calculus
of functions_, and_arithmetic_ the _calculus of values_.
It is easy to understand by this how the ordinary definitions are
insufficient and even vicious. More often than not, the exaggerated importance
accorded to signs has led to distinguish these two
fundamental branches of the science of calculus by the way of designating in
each the subjects of reasoning, which is obviously absurd in
principle and false in fact. Even the famous definition given by Newton
when he characterized algebra as universal arithmetic
certainly gives a very false idea of the nature of algebra and of
that of arithmetic [6].
[Note 6: I thought I should mention this
definition especially ; because it serves as a basis for the opinion that
many good minds, foreign to
mathematical science , are formed from the abstract part of this
science, without considering that at the time when this outline was
formed, the mathematical analysis was not sufficiently developed
for the general character proper to each of its
main parts to be properly grasped, which
explains why Newton was able to propose a definition then which
he would certainly reject today.]
After having established the fundamental division of the _calculus_ into two
main branches , I must compare, in general, the extent, the importance and the
difficulty of these two kinds of calculation, in order to have no more to
consider only the _calculus of functions_ , which must be the
essential subject of our study.
The _calculus of values_, or the_arithmetic_, seems, at first glance, to
have to present a field as vast as that of_algebra_,
since it seems to have to give rise to as many distinct questions as
one can conceive of different algebraic formulas to evaluate. But
a very simple reflection is enough to show that the domain of calculation
of values is, by its nature, infinitely less extensive than that of the
calculus of functions. Because, by distinguishing the functions in _simples_ and
_composées_, it is obvious that when one knows how to _evaluate_
simple functions , the consideration of the composite functions does not present any more, in
this respect, any difficulty. From the algebraic point of view, a
compound function plays a role very different from that of the
elementary functions which constitute it, and it is precisely from this that
all the principal analytical difficulties arise . But it is quite
different for arithmetic calculation. Thus, the number of
arithmetic operations , really distinct, is only marked by that of
elementary abstract functions, of which I have presented the
very little extended table above . The evaluation of these ten functions
necessarily gives that of all the functions, in infinite number, that we
consider in the whole of mathematical analysis, such, at least, as
it exists today. Whatever formulas that
the elaboration of the equations may lead to , there would be no place for new
arithmetic operations unless we came to create real
new analytical elements, the number of which will always be, whatever
happens, extremely small. The field of the_arithmetic_ is therefore, by its
nature, infinitely restricted, while that of the
It is important to note, however, that the field of _calculating values_
is, in reality, much larger than is
commonly imagined . For several questions, truly _arithmetic_,
since they consist in _evaluations_, are not
ordinarily classified as such, because we are in the habit of
treating them only as incidental, in the midst of a body of
more or less analytical research. lower: the too high opinion which is
commonly formed of the influence of signs is still the main cause of
this confusion of ideas. Thus, not only the construction of a
table of logarithms, but also the calculation of the trigonometric tables,
are true arithmetic operations of a superior kind. We
may also cite as being in the same case, although in a
very distinct and higher order, all the processes by which
the value of any function whatever is directly determined for each
particular system of values attributed to the quantities on which it depends,
when it is not possible to generally know the
explicit form of this function. From this point of view, the
_numeric_ resolution of the equations which we do not know how to solve _algebraically_,
and similarly the calculation of definite integrals of which we ignore the
general integrals, are really part, despite appearances, of the
field of_arithmetic_, in which it is necessary to
understand everything which has for object the_evaluation_ of functions. The
considerations relative to this goal, are in fact, constantly homogeneous,
whatever _evaluations_, and always quite distinct
from truly _algebraic_ considerations.
To complete the formation of a correct idea of the real extent of the calculus
of values, we must also understand that part of the
general science of calculus which today especially bears the name of
_theory of numbers_, and which is still so little advanced. This branch,
very extensive by its nature, but whose importance in the system
General of science is not very great, its object is to discover
the properties inherent in different numbers by virtue of their
values and independently of any particular numeration. It
therefore constitutes a sort of_transcendent arithmetic_; it is to this
that the definition proposed by Newton for
algebra would actually suit .
The total domain of_arithmetic_ is therefore, in reality, much larger
than is ordinarily conceived. But, nevertheless, whatever
legitimate development one may grant it, it remains certain
that, in the whole of abstract mathematics, the _calculus of
values_ will never be more than a point, so to speak, in comparison with the
_calculus of functions_, in which science essentially consists.
This appreciation will become even more sensitive by some
considerations which remain for me to indicate on the true nature of
arithmetic questions in general, when one examines them in a
detailed manner .
In seeking to determine exactly what
_evaluations_ consist of, it is easy to recognize that they are nothing
more than real _transformations_ of the functions to be evaluated,
transformations which, despite their special purpose, are nonetheless
essentially of the same nature as all those taught by
analysis. From this point of view, the _calculation of values_ could be
conceived simply as an appendix and a particular application of the
_calculus of functions_, so that the_arithmetic_ would
disappear, so to speak, in the whole of
abstract mathematics , as a separate section.
To fully understand this consideration, it must be observed that, when
we propose to_evaluate_ an unknown number whose mode of formation is
given, it is, by the very statement of the arithmetic question, already
defined and expressed under a certain form; and that by_evaluating_ it, we only
put our expression in another determined form, to
which we are accustomed to relate the exact notion of each number
in particular, by making it fit into the regular system of
_numeration_. The_evaluation_ consists so well in a simple
_transformation_, that when the primitive expression of the number is itself found
in conformity with the regular numeration, there is no more,
strictly speaking, of_evaluation_, or rather one responds to the question by
the question itself. If we ask, for example, to add the two numbers
thirty and seven, we will answer by limiting ourselves to repeating the very wording of the
question, and we will nevertheless believe to have _evaluated_ the sum, which
means that, in this case , the first expression of the function does not
need to be transformed; while it would not be so for
add twenty-three and fourteen, for then the sum would not be
immediately expressed in a manner consistent with the rank it occupies
in the fixed and general scale of numeration.
By specifying, as much as possible, the preceding consideration, we can
say that_evaluating_ a number is nothing more than putting its
primitive expression in the form / [a + b / beta + c / beta ^ 2 + d / beta ^ 3 + e / beta ^ 4
/ ldots + p / beta ^ m /] / beta being ordinarily equal to 10; and the
coefficients a, b, c, d, etc. being subject to these conditions of being
integers less than / beta, which can become zero, but never
negative. So any arithmetic question is likely to be
posited as consisting in putting in such a form
any abstract function of various quantities which one supposes themselves already to have
a similar form. We could therefore see in the
various operations of arithmetic only simple particular cases
of certain algebraic transformations, except for the
special difficulties arising from the conditions relating to the state of the coefficients.
It follows clearly from the above that abstract mathematics
consists essentially of the _calculus of functions_, which was
obviously already the most important, the most extensive, and the
most difficult part of it. This will henceforth be the exclusive subject of our
analytical considerations. So, without dwelling further on _calculating
values_, I will immediately move on to discussing the
fundamental division of _calculating functions_.
At the beginning of this lesson, we determined what
the real difficulty is in equating
mathematical questions. It is essentially because of
the insufficiency of the very small number of analytical elements which we
possess, that the relation of the concrete to the abstract is ordinarily so
difficult to establish. Let us now try to appreciate philosophically
the general process by which the human mind has arrived, in such a
large number of important cases, to overcome this fundamental obstacle.
By directly considering the whole of this capital question, one is
naturally led to conceive first of all a first means to
facilitate the establishment of the equations of the phenomena. Since the
main obstacle on this subject comes from the too few of our
analytical elements , everything would seem to boil down to creating new ones. But this approach
, however natural it may seem, is truly illusory
when we examine it in depth. Although it can
certainly be useful, it is easy to be convinced of its
necessary insufficiency.
Indeed, the creation of a real new abstract function
elementary presents, by itself, the greatest difficulties. There
is even, in such an idea, something which seems contradictory.
Because a new analytical element would obviously not
meet the essential conditions which are proper to it, if one could not
immediately_evaluate it: now, on the other hand, how to _evaluate_ a
new function which would be really _simple_, that is say, who
would not fit in a combination of those already known? It seems
almost impossible. The introduction into the analysis of another
elementary abstract function , or rather of another pair of functions (because
each would always be accompanied by its _inverse_), therefore supposes
necessarily the simultaneous creation of a new
arithmetic operation , which is certainly very difficult.
If we seek to form an idea of the means which the human mind
might employ to invent new analytical elements, by
examining the processes by which it has actually designed those
which we have, observation leaves us to This respect in
complete uncertainty, for the artifices which he has already used for
this are evidently exhausted. In order to convince ourselves of this, let us consider the
last couple of simple functions which were introduced in
the analysis, and whose formation we have, so to speak, assisted,
know the fourth couple, because, as I have explained, the fifth
couple does not, properly speaking, constitute real new
analytical elements. The function a ^ x, and, consequently, its inverse, were
formed by conceiving from a new point of view a function already
known for a long time, the powers, when the notion has been
sufficiently generalized. It was enough to consider a power
relative to the variation of the exponent, instead of thinking of the
variation of the base, for a
truly new simple function to result , the variation then following a completely
different course. But this trick, as simple as it is ingenious, can no longer
provide nothing. For by turning over all our
present analytical elements in the same way , we only end up making them fit into each
other.
We therefore do not in any way conceive of how we could proceed
to the creation of new elementary abstract functions,
suitably fulfilling all the necessary conditions. This does
not mean, however, that today we have reached the
effective limit set in this regard by the limits of our intelligence. It is
even certain that the last special improvements in
mathematical analysis have contributed to extending our resources in this respect, by
introducing into the domain of calculus certain definite integrals,
which, in some respects, take the place of new simple functions,
although they are far from fulfilling all the suitable conditions,
which has prevented me from inscribing them in the table of true
analytical elements . But, all things considered, I believe it remains
indisputable that the number of these elements can only increase with
extreme slowness. Thus, it cannot be in such a process that
the human mind has drawn its most powerful resources to
facilitate as much as possible the establishment of the equations.
This first means being ruled out, there is obviously only one left;
that is, given the impossibility of directly finding the equations between the
quantities that we consider, to look for corresponding ones between
other auxiliary quantities, linked to the first according to a certain
determined law, and from the relation of which we then go back to that
of the primitive quantities. Such is, in fact, the
eminently fruitful conception that the human mind has succeeded in founding, and which
constitutes its most admirable instrument for the mathematical exploration
of natural phenomena, the so-called "transcendent" analysis.
In general philosophical thesis, the auxiliary quantities which one
introduces, instead of the primitive quantities or concurrently with them,
to facilitate the establishment of the equations, could be derived according to
some law whatever of the immediate elements of the question. Thus, this
conception has much more significance than has been
commonly assumed by even the most profound surveyors. It is extremely important
to represent it in all its logical extent; because it is perhaps by
establishing a general mode of _dérivation_ other than that to which we
have constantly limited ourselves so far, although it is not, obviously,
the only possible one, that we will one day succeed in perfecting
essentially the whole of mathematical analysis, and consequently to
found, for the investigation of the laws of nature, even
more powerful means than our current procedures, susceptible, no doubt,
of exhaustion.
But, to have regard only to the present constitution of science,
the only auxiliary quantities usually introduced in place
of the primitive quantities in the_transcendent analysis_, are what are
called the _infinitely small_ elements, the _differentials_ of various
orders of these quantities, if we conceive of this analysis in the manner of
Leïbnitz; or the _fluxions_, the _limits_ of the ratios of the
simultaneous increases of the primitive quantities compared to each
other, or, more briefly, the _first_ and _last reasons_ of
these increases, adopting Newton's conception; or, finally,
the _dérivées_ proper of these quantities, that is to say, the
coefficients of the different terms of their respective increases,
according to the conception of Lagrange. These three main ways
of considering our current transcendent analysis, and all the others
less distinctly clear cut that have been proposed successively, are,
by their nature, necessarily identical, either in the calculation or
in the application, as I will explain it in general
in the sixth lesson. As to their relative value, we will then see
that Leïbnitz's conception has hitherto had
incontestable superiority in use , but that its logical character is eminently vicious;
while Lagrange's conception, admirable for its simplicity, for
its logical perfection, by the philosophical unity that it has established in
the whole of mathematical analysis, until then divided into two
almost independent worlds, still presents, in applications,
serious drawbacks, by slowing down the progress of intelligence:
Newton's conception more or less holds the middle in these various
reports, being less rapid, but more rational than that of
Leïbnitz, less philosophical, but more applicable than that of
Lagrange.
This is not the place to explain exactly how the
consideration of this kind of auxiliary quantities introduced into the
equations in place of the primitive quantities really facilitates
the analytical expression of the laws of phenomena. The sixth lesson will be
specially devoted to this important subject, considered from the
different general points of view to which
transcendent analysis has given rise . I limit myself for the moment to considering this conception in
the most general way, in order to deduce from it the fundamental division
of the _calculus of functions_ into two essentially distinct calculations,
including the concatenation, for the complete solution of the same
mathematical question. , is invariably determined.
In this respect, and in the rational order of ideas,
transcendent analysis presents itself as necessarily being the first,
since its general aim is to facilitate the establishment of the
equations, which must obviously precede the _resolution_ proper
of these equations, which is the object of ordinary analysis. But,
although it is eminently important to conceive of the true
sequence of these two analyzes in this way, it is nonetheless suitable, in
accordance with constant usage, to study transcendent analysis
only after ordinary analysis; for, if, at bottom, it is by
itself logically independent of it, or if, at least, it is possible
today to essentially free it from it, it is clear that its use
in the solution of questions always having more or less need
to be completed by that of ordinary analysis, one would be forced to
leave the questions in abeyance, if it had not been studied
beforehand.
As a result of the foregoing, the _calculus of functions_,
or_algebra_, taking this word in its greatest extension, consists
of two distinct fundamental branches, one of which has as its
immediate object the _resolution_ of the equations, when those - these are
immediately established between the same quantities that are considered; and of
which the other, starting from equations, much easier to form in
general, between quantities indirectly linked to those of the problem, has
for its own and constant purpose to deduce, by processes
invariable analytical, the corresponding equations between the
direct quantities which one considers, which makes return the question
in the field of the preceding calculation. The first computation carries, most
often, the name of_analysis ordinary_, or of_algebra_ proper;
the second constitutes what is called the_transcendent analysis_, which has
been designated by the various denominations of _finitesimal
calculation_ , _calcul des fluxions et des fluentes_, _calcul des vanissans_, etc.,
depending on the point of view designed. To rule out any
foreign consideration, I will propose to name it _calcul des
functions indirect_, by giving to ordinary analysis the title of
_calculation of direct functions_. These expressions, which I form
essentially by generalizing and clarifying Lagrange's ideas,
are intended to simply indicate with accuracy the true
general character proper to each of the two analyzes.
Having established the fundamental division of mathematical analysis, I must
now consider the whole of each of its two
parts separately , beginning with the _calculus of direct functions_, and
then reserving more extensive developments for the various branches
of the _calculus of indirect functions_ .
FIFTH LESSON.
SUMMARY. General considerations on the calculation of direct functions.
According to the general explanation which ends the preceding lesson, the
_calculus of direct functions_, or the_algebra_ properly so called, is
entirely sufficient for the solution of mathematical questions, when they are
simple enough so that we can immediately form the equations between
the same quantities which are considered, without it being necessary
to introduce in their place or jointly with them any system of
auxiliary quantities _derived_ from the first. In truth, in the
greatest number of important cases, its use needs to be preceded
and prepared by that of the _calculus of indirect functions_, intended to
facilitate the establishment of the equations. But although the role of
algebra is then only secondary, it nevertheless always has
a necessary part in the complete solution of the question, so
that the _calculus of direct functions_ must continue to be, by its
nature, the fundamental basis of any mathematical analysis. We
must therefore, before going any further, consider, in
general terms, the rational composition of this calculation, and the degree of
development to which it has reached today.
The definitive object of this calculation being the _resolution_ proper of the
_equations_, that is to say, the discovery of the mode of formation of the
unknown quantities by the quantities known from the _equations_
that exist between them; it naturally presents as many
different parts as one can conceive of truly
distinct classes of equations ; and consequently, its proper extent is rigorously
indefinite, the number of analytical functions capable of entering
into the equations, being by itself quite unlimited, although
they are composed only of a very small number of
primitive elements .
The rational classification of equations must evidently be
determined by the nature of the analytical elements of which their
members are composed ; any other classification would be essentially arbitrary.
In this respect, analysts first divide the equations to one or to
several variables in two main classes, depending on whether they only
contain functions of the first three pairs (_ see the
table, 4th lesson, page 173), or whether they also contain
functions, either exponential or circulars. The denominations of
_algebraic_ and _transcendent_ functions,
commonly given to these two principal groups of analytical elements, are,
without doubt, very unsuitable. But the division universally
established between the corresponding equations is none the less
very real, in the sense that the resolution of the equations containing the
so-called _transcendent_ functions necessarily presents more than
difficulties than those of the so-called _algebraic_ equations. Also the study
of the former has so far been excessively imperfect, to such an extent
that often the resolution of the simplest of them is
still unknown to us [7]; it is on the elaboration of the seconds that
our analytical methods relate almost exclusively.
[Note 7: However simple it may seem, for
example, the equation / [a ^ x + b ^ x = c ^ x, /] we do not
yet know how to _resolve_ it; which may give an idea of
the extreme imperfection of this part of the algebra.]
Considering now only these _algebraic_ equations, we must
first observe that, although
_ irrational_ functions of unknowns as well as
_rational_ functions ; one can always, by more or less
easy transformations , make the first case fit into the second; ensure that it is
the latter that analysts had to look only for
r solve all _algébriques_ equations.
In the infancy of algebra, these equations were classified according to
the number of their terms. But this classification was obviously
flawed; as separating really similar cases, and bringing together
others which had nothing in common but a character of no
real importance [8]. It has only been maintained for the equations with
two terms, susceptible, in fact, of a common resolution of their
own.
[Note 8: The same momentary error was made later
in the early stages of the calculus, for
the integration of differential equations.]
The classification of equations, according to what are called their
_degrés_, universally accepted for a long time - time by analysts,
is, on the contrary, eminently natural, and deserves to be pointed out here.
For, by comparing, in each _degre_, only the equations which
correspond to each other, as to their relative complication, we can say that
this distinction rigorously determines the difficulty more or less
great of their _resolution_. This gradation is
indeed sensitive , for all the equations that we know how to solve. But
this can be seen in general, regardless of the
fact of the resolution. It suffices, for this, to consider that
the most general equation of each degree necessarily includes
all those of the various lower degrees, so that it must be
so with the formula which determines the unknown. Consequently, however
weak one might suppose _a priori_ the difficulty proper to the _degree_ which
one considers, as it is inevitably complicated, in
the execution, of those presented by all the preceding _degrees_, the
So resolution really offers more obstacles as the degree
of the equation increases.
This increase in difficulty is such that until now the solution of
algebraic equations is known to us only in the first four
degrees only. In this regard, algebra has not made
considerable progress since the work of Descartes, and of the Italian analysts
of the sixteenth century, although, in the last two centuries, there has
perhaps not been a single geometer who not bothered to push
further before solving the equations. The general equation of the fifth
degree itself has so far withstood all attempts.
presenting the formulas to solve the equations as the degree
increases, the extreme embarrassment already caused by the use of the formula of the
fourth degree, and which makes it almost inapplicable, determined the
analysts to give up, by an agreement tacitly, to pursue similar
research, although they are far from considering it as impossible
ever to obtain the solution of the equations of the fifth degree, and of
several other higher degrees. The only question of this kind, which
would be really of great importance, at least
logically, would be the general resolution of algebraic equations of
any degree. However, the more we meditate on this subject, the more we are led to
to think, with Lagrange, that it really exceeds the effective range of
our intelligence. It should also be observed that the formula which would
express the _root_ of an equation of degree m should necessarily
contain radicals of the order m (or functions of an
equivalent multiplicity ), because of the m determinations which it must contain.
Since we have seen, moreover, that it must also embrace, as a
particular case , that which corresponds to any other lower degree, it
follows that it would inevitably contain radicals of
the order m- 1, others of the order m-2, etc., in such a way that, if it
were possible to discover it,
great complication to be able to be usefully employed, unless one
manages to simplify it, while preserving however all the
suitable generality, by the introduction of a new kind of
analytical elements , of which we have as yet no idea . There is therefore reason to
believe that, without having already reached in this respect the limits imposed
by the limited scope of our intelligence, we would not delay in
meeting them by prolonging this
series of research with strong and sustained activity .
It is also important to observe that, even supposing that the
resolution of the _algebraic_ equations of any degree has been obtained , we
would still have dealt only with
proper, that is to say, the calculation of direct functions,
embracing the resolution of all the equations that can form the
analytical functions known today. Finally, to complete the
elucidation of the philosophical consideration of this subject, it must be
recognized that, by an indisputable law of human nature, our
means for conceiving new questions being much more
powerful than our resources for solving them, or, in other words, In other words,
the human mind being much more apt to imagine than to reason, we
will necessarily always remain below difficulty, whatever
degree of development our intellectual labors ever reach.
Thus, even if one day we discover the complete resolution of
all the analytical equations currently known, which, on
examination, must be considered quite chimerical, there is no doubt that
before reaching This goal, and probably even as a
subsidiary means , would have already overcome the much less difficulty, though
very great, of conceiving new analytical elements,
the introduction of which would give rise to classes of equations which we
are completely ignorant of today. hui; so that such a
relative imperfection of algebraic science would recur again, in spite of
the real increase, very important moreover,
In the present state of algebra, the complete resolution of the equations
of the first four degrees, of any binomial equations, of
certain special equations of the higher degrees, and of a very small
number of exponential, logarithmic, or circular equations ,
therefore constitute the fundamental methods presented by the calculation of
direct functions for the solution of mathematical problems. But,
with such limited elements, geometers have none the less
succeeded in treating, in a truly admirable manner, a very large
number of important questions, as we shall recognize
successively in the remainder of this volume. General improvements
Introduced a century ago into the total system of
mathematical analysis, their main character was to use to an
immense degree this little knowledge acquired on the calculation of
direct functions , instead of tending to increase them. This result has been obtained to
such a point that most often this calculation has an effective role in the
complete solution of the various questions only through its
simplest parts , those which relate to the equations of the first two
degrees, at a alone or with several variables.
The extreme imperfection of algebra, relative to the solution of
equations, has determined analysts to deal with a new class
of questions, the true character of which must be emphasized here.
When they thought it necessary to give up any longer pursuing the
resolution of the algebraic equations of degrees greater than the fourth,
they took care to make up, as far as possible, for this immense
gap, by what they called the _numerical resolution_ of the equations.
Not being able to obtain, in most cases, the _ formula_ which expresses
which explicit function the unknown is data, we sought, in the
absence of this resolution, the only truly _algebraic_, to
determine, at least, independently of this formula, the _value_ of
each unknown for such or such designated system of particular values
assigned to the data. Through the successive works of analysts, this
incomplete and bastard operation, which presents an intimate mixture of
truly algebraic questions with purely
arithmetic questions , has been able, at least, to be entirely carried out in all
cases, for equations of one degree and even of any shape. In
this respect, the methods which we have today are sufficiently
general, although the calculations to which they lead are often
almost inexecutable, on account of their complication. In
this regard, therefore , all that remains is to simplify the procedures enough so that they
become regularly applicable, which we can hope to obtain.
in the following. According to this state of the computation of direct functions, one
then endeavors, in the application of this computation, to arrange, as much
as possible, the questions proposed in such a way as ultimately to require only
this _numeric_ resolution of the equations.
However precious such a resource may be, in the absence of
a true solution, it is essential not to ignore the true
character of these procedures, which analysts rightly regard
as a very imperfect algebra. Indeed, it is far from being
able to always reduce our mathematical questions to
depend, in the final analysis, only on the _numerical_ resolution of the
equations. This can only be done for quite isolated questions,
or truly final, that is to say, for the smallest number. Most
of the questions are in fact only preparatory, and intended to
serve as an indispensable preliminary to the solution of other questions.
Now, for such a goal, it is obvious that it is not the
actual _value_ of the unknown that it is important to discover, but the _ formula_
which shows how it derives from the other quantities considered. This is
what happens, for example, in a very extensive case, whenever
a given question simultaneously contains several unknowns.
It is then, as we know, to make it, above all, the separation.
By properly employing, for this purpose, the simple and general process
fortunately imagined by analysts, and which consists in relating
one of the unknowns to all the others, the difficulty would
constantly disappear , if one always knew how to solve the
equations considered algebraically. , without the _numeric_ resolution being
of any use. It is only for lack of knowing the
_algebraic_ solution of equations with a single unknown, that we are
obliged to treat_elimination_ as a separate matter, which
forms one of the greatest special difficulties of ordinary algebra.
However painful the methods by which one overcomes
This difficulty, they are not even applicable in an
entirely general way , to the elimination of an unknown factor between two
equations of any form.
In the simplest questions, and when we really only have to
solve a single equation with a single unknown, this
_numeric_ resolution is none the less a very imperfect process, even when
it is strictly sufficient. It has, in fact, the serious
drawback of forcing to redo the entire sequence of operations for the
slightest change which may occur in only one of the quantities
considered, although their relation always remains the same, without the
calculations made for one case can in no way dispense with those
which concern another very little different case, for want of having been able to
abstract and treat distinctly this purely algebraic part of
the question which is common to all the cases resulting from the simple
variation given numbers.
According to the preceding considerations, the computation of
direct functions , considered in its current state, is therefore naturally divided into
two very distinct parts, depending on whether one deals with the
_algebraic_ resolution of the equations or their _numeric_ resolution. The
first part, the only one really satisfactory, is unfortunately
very little extended, and will probably always remain very limited; the
the second, most often insufficient, at least has the advantage of a
much greater generality. The necessity of clearly distinguishing
these two parts is obvious, because of the essentially different aim
which one proposes oneself in each, and consequently, of the proper point of view under
which one considers the quantities there. Moreover, if we consider them
relatively to the various methods of which each is composed, we find
in their rational distribution an entirely different course. In
fact, the first part should be divided according to the nature of the
equations we can solve, and independently of any
consideration relating to the unknown _valeurs_. In the second
In part, on the contrary, it is not according to the _degrees_ of the equations
that the processes are naturally distinguished, since they are
applicable to equations of any degree; it is according to the
numerical species of the _values_ of the unknowns. Because, to calculate
these numbers directly without deducing them from the formulas which would make known the
expressions, the means could obviously not be the same, when the
numbers are susceptible to be evaluated only by a series
of approximations always incomplete, only when 'we can get them
exactly. This distinction so important, in the
numerical resolution of equations, of immeasurable roots, and of roots
commensurables, which require quite different principles for
their determination, is entirely insignificant in
algebraic resolution , where the _rational_ or _irrational_ nature of the numbers
obtained is a simple accident of calculation, which can exert no
influence on the methods employed. . It is, in short, a simple
arithmetic consideration. The same can be said, albeit to a lesser
degree, of the distinction of commensurable roots themselves into
whole and fractional ones. Finally, it is also the same, with a fortiori
, for the most general classification of roots, into
_real_ and _imaginaire_. All these various considerations, which are
predominant in the numerical resolution of the equations, and which
have no importance in the algebraic resolution, make more
and more sensitive the essentially distinct nature of these two
principal parts of algebra proper.
These two parts, which constitute the immediate object of the calculation of
direct functions, are dominated by a third purely
speculative, from which both borrow their
most powerful resources , and which has been very exactly designated by the
general name of _theory of equations_, although however it
still only relates to so-called _algebraic_ equations. The resolution
numerical equations, because of their generality, especially require
this rational basis.
This last branch of algebra, which is so important, is
naturally divided into two orders of questions, first those which
relate to the composition of equations, and then those which
concern their transformation; the latter having for object to
modify the roots of an equation without knowing them, according to
any given law , provided that this law is uniform relative to
all these roots [9].
[Note 9: I think I should, on the subject of the theory of
equations, point out here a gap of some importance. The
fundamental principle on which it rests, and which is so
frequently applied in all mathematical analysis, the
decomposition of algebraic, rational, and
integer functions, of any degree, into factors of the first
degree, is never employed except for functions of a
single variable, without anyone having considered whether it should be
extended to functions of several variables, which
nevertheless should not be left uncertain. As for the
functions of two or three variables,
geometric considerations clearly decide, albeit in an
indirect way , that their decomposition into factors is
ordinarily impossible; because it would follow from it that each
corresponding class of equations could not represent
a line or a surface _sui generis_, and that its
geometrical locus would always enter into the system of those
belonging to equations of lower degree,
so that, gradually close, any equation would
never produce anything but straight lines or planes. But,
precisely because of this concrete interpretation, this
theorem, although purely negative, seems to me to have such
great importance for analytical geometry that I
am astonished that no attempt has been made to establish a direct
also characteristic difference between functions with
only one variable and those with several variables. I will
report here briefly the abstract and
general demonstration that I found of it, although it was more
suitably placed in a special treatise.
1º If f (x, y) could be decomposed into factors of the first
degree, we would obtain them by solving the equation f (x, y) = 0.
Now, according to the considerations indicated in the text,
this equation, solved with respect to x, would provide
formulas which would necessarily contain various radicals,
in which y would enter. The functions of y, enclosed
under each radical, obviously cannot in general be
perfect powers. Now, they would have to
guess this for the elementary factors corresponding
to f (x, y), and which are already of the first degree in x, to be
also of the first degree, or even simply rational,
relatively to y. This can therefore only take place in
certain particular cases, when the coefficients
fulfill the more or less numerous, but
constantly determined, conditions required by the disappearance of the
radicals. The same reasoning would obviously apply, with
much more reason, to the functions of three, four, etc.
variables.
2. Another demonstration, of a very different nature, is
drawn from the measure of the degree of generality of functions with
several variables, which is estimated by the number of
arbitrary constants entering into their most
complete and simple expression . I will confine myself to indicating it for
the functions of two variables; it would be easy to extend it
to those which contain more.
We know that the number of arbitrary constants contained
in the general formula of a function of degree m with two
variables is / frac {m (m + 3)} {2}. Now, if such a function
could only be decomposed into two factors, one of
degree n, and the other of degree mn, the product would contain a
number of arbitrary constants equal to / [/ frac {n (n + 3)} {2} +
/ frac {(mn) (m-n + 3)} {2} ./] This number being, as it is easy
to see, lower than the preceding one of n (mn), it follows
that such a product, having less generality than the
primitive function , cannot represent it constantly. We even see
that such a comparison would require n (mn)
special relations between the coefficients of this function, which one
would easily find by developing the identity.
This new kind of demonstration, based on a
consideration ordinarily neglected, could probably
be employed with advantage in several other
circumstances.]
To complete this rapid general enumeration of the various
essential parts of the calculus of direct functions, I must finally
expressly mention one of the most fruitful and important theories
of algebra proper, that relating to the transformation of
functions into series using what is called the method of
indeterminate coefficients. This method, so eminently analytical, and
which must be regarded as one of the most remarkable discoveries
of Descartes, has undoubtedly lost its importance since the invention and
the development of the infinitesimal calculus, of which it could take place
so happily in some particular respects. But the
growing extension of transcendent analysis, although having made this method
much less necessary, has, on the other hand, multiplied its applications
and increased its resources; so that by the useful combination which
finally took place between the two theories, the use of the method
of indeterminate coefficients has become today much more extensive
than it was even before the formation of the calculus of
indirect functions .
After having sketched the general picture of algebra proper,
main points of the calculation of direct functions, the notions of which
can be usefully clarified by a philosophical examination.
The difficulties relating to several singular symbols to which
algebraic calculations lead, and in particular to so-called
imaginary expressions , have been, it seems to me, greatly exaggerated as a result of
purely methaphysical considerations that we have tried to
introduce, instead of view these anomalous results from their true
point of view, as mere analytical facts. By conceiving them
thus, it is easy to recognize, as a general thesis, that the spirit of
mathematical analysis consisting in considering magnitudes under the
Only from the point of view of their relations, and independently of any idea of
determined value, it necessarily results for the analysts
the constant obligation to admit indifferently all kinds
of expressions whatever which
algebraic combinations can generate . If they wanted to refrain from a single one, because of its
apparent singularity, as it is always likely to occur
according to certain particular assumptions about the values
of the quantities considered, they would be forced to alter the
generality of their conceptions. , and by thus introducing into each
reasoning a series of truly foreign distinctions, they
would cause mathematical analysis to lose its main
characteristic advantage , the simplicity and uniformity of the ideas which it
combines. The embarrassment which the intelligence ordinarily experiences with regard to
these singular expressions, seems to me to come essentially from the
vicious confusion which it makes in its unsuspecting between the idea of _function_
and the idea of _value_, or, which amounts to the even, between the
_algebraic_ point of view , and the _arithmetic_ point of view. If the nature of this
work allowed me to present
sufficient developments in this regard , it would be easy for me, I believe, by a suitable use of the
considerations indicated in this lesson and in the two preceding ones,
to dispel the clouds in which a false way of seeing
usually surrounds these various notions. The result of this examination
would expressly demonstrate that mathematical analysis is, by its nature,
much clearer, under the various reports I have just
spoken of, than is commonly believed by geometers themselves, led astray
by the vicious objections of metaphysicians. .
With regard to negative quantities, which, as a result of the same
metaphysical spirit , have given rise to so many inappropriate discussions, as
devoid of any rational foundation as devoid of any real
scientific utility, we must distinguish, always considering the
simple analytical fact, between their abstract meaning and their
concrete interpretation, which we have almost always confused until
now. In the first respect, the theory of negative quantities
can be completely established by a single algebraic view.
As for the need to admit this kind of results concurrently with
any other, it derives from the general consideration which I have just
presented: and as for their use as an analytical artifice to make
the formulas more extensive, this calculation mechanism cannot really
give rise to no serious difficulty. Thus, we can consider the
abstract theory of negative quantities as leaving nothing
essential to be desired: it only really presents obstacles that
are inappropriately introduced into it by sophistic considerations.
But, it is by no means the same for their concrete theory.
From this point of view, it essentially consists in this admirable
property of the + and - signs of representing analytically the
oppositions of meaning of which certain quantities are susceptible. This
general theorem on the relations of the concrete to the abstract in
mathematics is one of the most beautiful discoveries that we owe to the
genius of Descartes, who obtained it as a simple result of
properly directed philosophical observation. A large number of
geometers have since attempted to establish a
general demonstration directly . But so far their efforts have been illusory, whether they
have tried to settle the difficulty by vain
metaphysical considerations , or by very risky comparisons, or because they have
taken simple verifications in some particular case more or less.
bounded for real demonstrations. These various
vicious attempts , and the heterogeneous mixture of the abstract point of view with the
concrete point of view, have even commonly introduced in this respect
such confusion, that it becomes necessary to state here distinctly the
general fact, whether one just want to use it, either
that we propose to explain. It consists, independently of any
explanation, in that: if in any equation expressing the
relation of certain quantities susceptible of opposing meaning, one
or more of these quantities come to be counted in a direction
contrary to that which they affected when the equation was
originally established; it will not be necessary to directly form
a new equation for this second state of the phenomenon; it will suffice to
change, in the first equation, the sign of each of the quantities
which will have changed direction, and the equation thus modified will
always coincide rigorously with that which would have been found by starting over
to seek for this new case the analytical law of the phenomenon. It is
in this constant and necessary coincidence that the
general theorem consists . However, so far we have not really succeeded in realizing
this directly; this has only been ascertained by a large number of
geometrical and mechanical verifications, which are, it is true, sufficiently
multiplied and above all sufficiently varied so that there cannot remain in
any just mind the slightest doubt about the correctness and the generality of
this essential property, but which, from a philosophical point of view, in no
way dispenses with seeking such an important explanation.
The extreme scope of the theorem must make it understood both and the
capital difficulty of this research so often
unsuccessfully repeated , and the high utility which would undoubtedly be, for the
improvement of mathematical science, the general conception of
this great truth, the spirit being obviously able to rise to it only by
placing itself to a point of view from which he would inevitably discover
new ideas, by the direct and deep consideration of the
relation of the concrete to the abstract. Be that as it may, the imperfection that
science still presents in this respect has not prevented
geometers from making the most extensive and important use of this
property in all parts of concrete mathematics, where one
feels an almost continual need for it. One can even derive
some logical utility from the simple clear consideration of this
general fact , as I have described it above; it follows, for example,
independently of any demonstration, that the property of which we
speak must never be applied to the quantities which affect
continuously variable directions, without giving rise to a simple
opposition of meaning: in this case, the sign of which is is necessarily
affected any result of calculation is not susceptible of any
concrete interpretation, and it is wrong that we sometimes try
to establish it; this circumstance takes place, among other occasions, for
vector rays in geometry, and for divergent forces in
mechanics.
A second general theorem on the relation of the concrete to the abstract in
mathematics, which I believe should be expressly considered here, is that
which is commonly referred to as the principle of_homogeneity_.
It is undoubtedly much less important in its applications than the
preceding one. But it particularly deserves our attention, as having,
by its nature, an even greater extent, since it applies
indiscriminately to all phenomena, and because of the real utility
which is often derived from it for the verification of their analytical laws.
I can moreover give a direct and general demonstration of it, which
seems to me very simple. It is based on this single observation,
self-evident: the accuracy of any relation between
any concrete magnitudes is independent of the value of the
_units_ to which they are referred to express them in numbers. For
example, the relation which exists between the three sides of a
right triangle , takes place either that one evaluates them in meters, or in leagues, or in
inches, etc.
It follows from this general consideration, that any equation which expresses
the analytical law of any phenomenon, must enjoy this
property of not being altered in any way, when simultaneously subjected to
to all the quantities therein, the change corresponding to
that which their respective units would experience. Now, this change
obviously consists in that all the quantities of each species would
become at the same time m times smaller, if the unit which
corresponds to them becomes m times greater, or vice versa. Thus, any
equation which represents any concrete relation, must have the
characteristic of remaining the same, when
all the quantities that it contains are made m times greater therein , and which express the magnitudes
between which the relation exists, excepting however the numbers
which simply designate the mutual _reports_ of these various
quantities, which remain invariable in the change of units.
It is in this property that the law of homogeneity consists, according to
its broadest acceptation, that is to say, of some
analytical functions that the equations are composed.
But, most often, we only consider the cases where these functions are
those which are particularly called _algebraic_, and to which the
notion of _degre_ is applicable. In this case, we can
further specify the general proposition, by determining the
analytical character that the equation must necessarily present for this
property to be verified. It is easy to see then, in fact, that by
the modification explained above, all the _terms_ of the first degree,
whatever their form, rational or irrational, whole or
fractional, will become m times larger; all those of the second
degree, m ^ 2 times; those of the third, m ^ 3 times, etc. Thus, terms of the
same degree, however diverse their composition may be, varying in
the same manner, and terms of different degrees varying in
unequal proportion, whatever similarity their
composition may offer , it will necessarily be necessary for the equation is not
confused, that all the terms it contains are of the same degree.
This is what the ordinary theorem of
the_homogeneity_; and it is from this circumstance that the general law
derived its name, which however ceases to be exactly suitable for
all other kinds of functions.
In order to deal with this subject in all its extent, it is important to observe
an essential condition, which must be taken into account in applying
this property, when the phenomenon expressed by the equation presents
magnitudes of various kinds. Indeed, it may happen that the
respective units are completely independent of each
other, and then the homogeneity theorem will take place, either with
respect to all the corresponding classes of quantities, or that we do not
want to consider only one or more of them. But it
will happen on other occasions that the various units will have
obligatory relations with each other, determined by the nature of the question.
Then, it will be necessary to have regard to this subordination of the units in the
verification of the homogeneity, which will no longer exist in a purely
algebraic sense , and whose precise mode will vary according to the genre of the
phenomena. Thus, for example, to fix the ideas, when we
consider in the analytical expression of geometric phenomena, at
the same time lines, areas, and volumes, it will be necessary to observe that
the three corresponding units, are necessarily linked to each other,
so that, following the generally established subordination in this
regard, when the first becomes m times as large, the second
becomes m ^ 2 times, and the third m ^ 3 times. It is with such a
modification that homogeneity will exist in the equations, where we
must then, if they are _algebraic_, estimate the degree of each
term, by doubling the exponents of the factors which correspond to
areas, and tripling those factors relating to volumes [10].
These are the main general considerations,
doubtless very insufficient , but to which I am constrained to reduce myself by the
natural limits of this course, relative to the calculation of functions.
direct. We must now pass to the philosophical examination of the
calculus of indirect functions, the much
greater importance and extent of which demands a greater development.
[Note 10: I was led, twelve years ago, by my
daily teaching of mathematical science, to
construct this general theory of homogeneity. I
have since found that M. Fourier, in his great work on
heat, published in 1822, had, for his part, followed an
essentially similar course. Despite this happy
coincidence, which must naturally have been determined by the
direct consideration of such a simple subject, I have not
thought it right here to refer to its demonstration; the one which I
have just exposed having for principal object to embrace
the whole of the question, without regard to any
special application .]
SIXTH LESSON.
SUMMARY. Comparative exposition of the various general points of view from
which the calculation of indirect functions can be considered.
We have determined, in the fourth lesson, the
philosophical character proper to transcendent analysis, however one
can conceive of it, by considering only the general nature of its
effective destination in the whole of mathematical science. This
As we know, this analysis has been presented by geometers from several
points of view that are really distinct, although necessarily equivalent,
and always leading to identical results. We can reduce them to
three main ones, those of Leïbnitz, Newton and Lagrange, of which all
the others are only secondary modifications. In the
present state of science, each of these three general conceptions offers
essential advantages which belong exclusively to it, without having
yet succeeded in constructing a single method combining
all these various characteristic qualities. In meditating on
the whole of this great question, one is convinced, I believe, that
It is in Lagrange's conception that this
combination will one day take place . When this important philosophical work, which requires a
deep elaboration of all the fundamental mathematical ideas,
will be properly carried out; in order to know
the transcendent analysis, we will be able to limit ourselves then to the sole study of this
definitive conception ; the others are essentially of
historical interest only . But until that time, science will have to be considered,
in this respect, as being in a real provisional state, which
absolutely requires, even for the dogmatic exposition of this analysis, the
simultaneous consideration of the various general modes proper to the calculation of
indirect functions. However unsatisfactory as
this multiplicity of conceptions of an
always identical subject may seem from a logical point of view , it is certain that, without this indispensable
condition, one could only form today a
very insufficient notion of this subject. analysis, either in itself, or especially
relative to its applications, whatever the single mode that one
would have thought it necessary to choose. This lack of systematization in the
most important part of mathematical analysis will not appear at all
strange, if we consider, on the one hand, its extreme extent, its
superior difficulty, and on the other hand, its almost
recent. The generation of geometers has scarcely been renewed since the
primitive production of the conception, doubtless destined to coordinate
science, so as to give it a fixed and uniform character;
thus, intellectual habits have not yet
been sufficiently formed in this respect .
If it was a question here of tracing the reasoned history of the
successive formation of transcendent analysis, it would first be necessary to
distinguish carefully from the calculation of the indirect functions properly speaking,
the mother idea of the infinitesimal method, which can be conceived of. by
itself, independently of any calculation. We would see, therefore, that
the first germ of this idea, is already found in the constant process,
employed by the Greek geometers, under the name of _method of exhaustion_,
to pass from what is relative to straight lines to what concerns
curved lines, and which consisted essentially in substituting for the
curve the auxiliary consideration of an inscribed or circumscribed polygon,
according to which one rose to the curve itself, by
appropriately taking the limits of the primitive relations.
However incontestable this lineage of ideas may be, we would
give it a very exaggerated importance, seeing in this method
of exhaustion, the real equivalent of our modern methods, as
made several surveyors. For the ancients had no
rational and general means for determining these limits, which
usually constituted the greatest difficulty of the question; in
so their solutions were not subject to rules
abstract and unchanging, the uniform application had to lead with
certainty to the required knowledge, which is the main character
of our transcendental analysis. In short, it remained to generalize the
conception employed by the ancients, and above all, by considering it in a
purely abstract manner, to reduce it to calculation, which was
impossible for them. The first idea that was produced in this news
direction, really goes back to our great geometer Fermat, whom
Lagrange rightly presented as having sketched out the direct formation
of transcendent analysis, by his method for the determination of
_maxima_ and _minima_, and for the search for tangents, which consisted
essentially of Indeed, in introducing the auxiliary consideration of the
correlative increases of the proposed variables, increases
afterwards suppressed as zero, after the equations had undergone
certain suitable transformations. But, although Fermat was the
first to conceive of this analysis in a truly abstract manner, it was
still far from being regularly formed into a general and
distinct, having its own notation, and above all freed from the
superfluous consideration of terms, which ended up no longer being
counted in Fermat's analysis, after having nevertheless singularly
complicated by their presence all the operations. This is what
Leibnitz did so happily half a century later, after some
intermediate modifications made by Wallis, and especially by
Barrow, to Fermat's ideas; and thereby he was the true creator of
transcendent analysis as we use it today. This
momentous discovery was so ripe, like all the great
conceptions of the human mind at the time of their manifestation, that
Newton, on his side, had arrived at the same time, or a little before,
at an exactly equivalent method, considering this analysis from
a very different point of view, and which, although more rational in
itself, is really less suitable for giving to the
common fundamental method all the scope and ease given to it by
the ideas of Leïbnitz. Finally, Lagrange, setting aside the
heterogeneous considerations which had guided Leïbnitz and Newton,
later succeeded in reducing the transcendent analysis, in its greatest
perfection, to a purely algebraic system, which still lacks
more aptitude for applications. .
After this blow knowing summary on the general history of the analysis
transcendent, proceed to the dogmatic exposition of the three designs
main order to assess their properties exactly
characteristics, and find the necessary identity of methods
in drift. Let us start with that of Leïbnitz.
It consists, as we know, in introducing into the calculation, to
facilitate the establishment of the equations, the infinitely small elements of
which we consider as composed the quantities between which we
seek relations. These elements or _differentials_ will have
constant and necessarily simpler and easier relations between them.
to discover that those of the primitive quantities, and from which
one could then, by a special calculation having as its own destination
the elimination of these auxiliary infinitesimals, go back to the
sought equations , which it would have been most often impossible to obtain
directly. This indirect analysis can be done to varying degrees;
for, if we sometimes find too much difficulty in immediately forming
the equation between the very differentials of the quantities that we are
considering, it will be necessary, by a redoubled use of the same general artifice, to
treat, in their turn, these differentials as new
primitive quantities , and seek the relation between their elements infinitely
small, which, in relation to the final objects of the question, will be
the _differential seconds_; and so on, the same
transformation being able to be repeated any number of times, on the
condition always of finally eliminating the increasing number of the
infinitesimal quantities introduced as auxiliaries.
A mind still foreign to these considerations does not immediately perceive
how the use of these auxiliary quantities can
facilitate the discovery of the analytical laws of phenomena; for the
infinitely small increases in the magnitudes proposed being of the same
kind as they, their relations do not appear to have to be obtained more
easily, the more or less small value of a quantity not being able, in
fact, to exert any influence on a research necessarily
independent, by its nature, of any idea of value. But it is easy,
nevertheless, to explain very clearly, and in a quite
general manner, to what point, by such an artifice, the question must be
simplified. To do this, we must begin by distinguishing the
different orders of infinitely small, of which we can get a
very precise idea , by considering that they are either the successive powers
of the same infinitely small primitive, or quantities that we can
present as having finite relations with these powers, so
that, for example, the second, third, etc., differentials of the
same variable are classified as infinitely small of the second order, of the
third, etc., because it is easy to show in them multiples
finite powers second, third, etc., of a certain
first differential. These preliminary notions being posed,
the spirit of infinitesimal analysis consists in constantly neglecting
infinitely small quantities with respect to finite quantities, and,
generally, the infinitely small of any order with respect to all
those d 'a lower order. We can immediately see how much such a
faculty must facilitate the formation of equations between
differentials of quantities, since, instead of these differentials,
we can substitute such other elements as we wish, and which would
be simpler to consider, by conforming to this one condition, that
the new elements differ from the previous ones only in quantities
infinitely small compared to them. It is thus that it will be possible,
in geometry, to treat curved lines as composed of an
infinity of rectilinear elements, curved surfaces as formed
of plane elements; and, in mechanics, the motions varied as an
infinite series of uniform motions, succeeding each other at
infinitely small intervals of time . In view of the importance of this admirable design, I
believe it duty here, by the summary indication of some
principal examples , to complete to clear up its fundamental character.
Whether it is a question of determining, at each point of a plane curve whose
equation is given, the direction of its tangent, a question of which the
general solution was the primitive object that the
inventors of the transcendent analysis. We will consider the tangent as
a secant which would join two infinitely neighboring points; and then, by
naming dy and dx the infinitely small differences of the coordinates of
these two points, the first elements of the geometry will
immediately provide the equation t = / frac {dy} {dx}, for the tangent
trigonometric of the angle made with the x axis by the tangent
sought, which, in a rectilinear coordinate system, is the
easiest way to fix its position. This equation, common to
all the curves, being posed, the question is reduced to a simple
analytical problem, which will consist in eliminating the infinitesimals dx and
dy, introduced as auxiliaries, by determining, in each
particular case , according to the equation of the curve proposed, the ratio of dy
to dx, which will be done constantly by uniform and
very simple methods .
Second, we want to know the length of the arc of
any curve, considered as a function of the coordinates of its
ends. It would be impossible to immediately establish the equation
between this arc s and these coordinates, while it is easy to find the
corresponding relation between the differentials of these various
magnitudes. The simplest theorems of elementary geometry
will, in fact, immediately give, by considering the infinitely small arc
ds as a straight line, the equations / [ds ^ 2 = dy ^ 2 + dx ^ 2,
/ mbox {or} ds ^ 2 = dx ^ 2 + dy ^ 2 + dz ^ 2, /] depending on whether the curve will be flat
or with double curvature. In either case, the question is
now entirely within the domain of analysis, which will trace,
according to this relation, to that which exists between finite quantities
themselves which one considers, by the elimination of the differentials,
which is the proper object of the calculation of the indirect functions.
It would be the same for the squaring of curvilinear areas. If the
curve is plane and referred to rectilinear coordinates, we will conceive
the area A between it, the x-axis, and two extreme coordinates,
as increasing by an infinitely small quantity dA, as a result of a
similar increase of the abscissa. Then the relation between these two
differentials can be obtained immediately with the greatest
ease, by substituting for the curvilinear element of the proposed area the
rectangle formed by the extreme ordinate and the element of the abscissa, of which
it obviously differs only by an infinitely small quantity of the second
order, which will immediately provide, whatever the curve, the
very simple differential equation / [dA = ydx, /] hence the calculation of the
indirect functions , when the curve is defined, will learn to deduce the
finite equation , immediate object of the problem.
Similarly, in dynamics, when we want to know the expression of the
speed acquired at each instant by a body animated by a movement varied
according to any law, we will consider the movement as uniform
during the duration of an infinitely small element of time t, and we will
thus immediately form the differential equation of = vdt, v denoting the
speed acquired when the body has traversed space e, and from there it will be
easy to conclude, by simple invariable analytical procedures, the
formula which would give the speed in each particular movement,
according to the corresponding relation between time and l 'space; or,
conversely, what would this relation be if the mode of variation of
the speed were supposed to be known, either with respect to space or with
respect to time.
Finally, to indicate another kind of question, it is by a
similar course that, in the study of thermological phenomena, as
M. Fourier has so happily conceived, we can form very simply, thus
that we will see later, the general differential equation which
expresses the variable distribution of heat in any body whatever to
some influences that it is supposed to be subjected, according to the only relation,
very easy to obtain, which represents the distribution uniformity of
heat in a rectangular parallelepiped,
geometrically considering any other body as decomposed into infinitely
small elements of such a shape, and thermologically the heat flux as
constant for an infinitely small time. Consequently, all the
questions which abstract thermology can present will be
reduced, as for geometry and mechanics, to pure
difficulties of analysis, which will always consist in the elimination of the
differentials introduced as auxiliaries to facilitate
the establishment of the equations.
Examples of such a diverse nature are more than sufficient to make
clearly understood in general the immense scope of the
fundamental conception of transcendent analysis, as Leïbnitz formed it,
and which undoubtedly constitutes the highest thought to which
the human spirit has never been elevated until now.
We see that this conception was essential to complete the founding of
mathematical science, by making it possible to establish in a broad and
fruitful manner the relation of the concrete to the abstract.
be considered as the necessary complement of
Descartes' great mother idea , on the general analytical representation of
natural phenomena , an idea which only began to be worthily appreciated and
properly exploited since the formation of the
infinitesimal analysis , without which it could not yet produce, even in
geometry, very important results [11].
[Note 11: It is indeed quite remarkable that
men such as Pascal, paid so little attention to
Descartes' fundamental conception, without foreseeing
the general revolution which it was
necessarily destined to produce in the system. integer of
mathematical science. This came about because, without the
help of transcendent analysis, this admirable method
could not really yet lead to
essential results , which could not be obtained almost as well
by the geometric method of the ancients. The same minds
the most eminent always enjoyed far less
general methods by their very nature
philosophical, as the actual knowledge they
could get away.]
Even though I thought it necessary, in the preceding considerations, insist
particularly on the admirable facility that its nature presents
transcendent analysis for the search for the mathematical laws of all
phenomena, I must not neglect to bring out a second
fundamental property, perhaps as important as the first, and
which is no less inherent to it: I mean the extreme
generality of differential formulas, which express in a single
equation each determined phenomenon, however varied
the subjects in which it is considered may be. Thus, from the point of view of
infinitesimal analysis, we see, in the preceding examples, a
single differential equation giving the tangents to all the curves,
another their rectifications, a third their quadratures; and of
even, an invariable formula expressing the mathematical law of all
varied motion; finally a single equation constantly representing the
distribution of heat in a body and for any case.
This eminently remarkable generality, and which is for
geometers the basis of the highest considerations, is a happy
necessary and almost
immediate consequence of the very spirit of transcendent analysis , especially in Leibnitz's conception. It results from the
fact that, by substituting for the infinitely small elements of the quantities
considered, other simpler infinitesimals, which alone enter
into the differential equations, these infinitesimals are found, by
their nature, to be constantly the same for each total class of
questions, whatever the various objects of the phenomenon studied.
Thus, for example, any curve, whatever it is, being always
decomposed into rectilinear elements, one conceives _a priori_ that the relation
between these uniform elements must necessarily be the same for the
same unspecified geometrical phenomenon, although the
corresponding finite equation to this differential law must vary from one curve to
another. It is obviously the same in any other case.
The infinitesimal analysis has therefore not only provided a
general process for indirectly forming equations that would have been impossible.
to discover in a direct way; it has also made it possible to
consider, for the mathematical study of natural phenomena, a
new order of more general laws which nevertheless offer a
clear and precise meaning to any mind accustomed to their interpretation. These laws
are constantly the same for each phenomenon, in some objects
that we study, and only change in passing from one phenomenon to another;
hence it has been possible, moreover, by comparing these variations, to rise
sometimes, by a still more general view, to
positive comparisons between various classes of completely different phenomena,
according to the analogies presented by the differential expressions of
their mathematical laws. In the philosophical study of
concrete mathematics , I will endeavor to give an exact appreciation of this second
property characteristic of transcendent analysis, no less
admirable than the first, and by virtue of which the entire system
of an immense science, like geometry or mechanics, could be
found condensed in a small number of analytical formulas, from which
the human mind can deduce, by certain and invariable rules,
the solution of all the particular problems.
To end the general exposition of Leïbnitz's conception, it
remains for me now to consider in itself the demonstration of the process.
logic to which it leads, which unfortunately constitutes
the most imperfect part of this beautiful method.
In the early days of infinitesimal analysis, the
most famous geometers , such as the two illustrious brothers Jean and Jacques
Bernouilli, rightly attached much more importance to extending,
by developing it, the immortal discovery of Leïbnitz, and to
multiply its applications, to rigorously establish the
logical bases on which the methods of this new
calculation were based [12]. For a long time they contented themselves with responding with the
unexpected solution to the most difficult problems facing the opposition.
pronounced by most second-rate geometers against the
principles of the new analysis, doubtless convinced, contrary
to ordinary habits, that in mathematical science much more
than in any other, one can boldly welcome new
means, even when their rationality is imperfect, provided they
are fruitful, since, the verifications being much easier and
more multiplied, the error cannot remain long unnoticed.
Nevertheless, after the first impetus, it was impossible to stop there; and
it was necessarily necessary to return to the very foundations of the
Leïbnitzian analysis in order to note generally the rigorous accuracy of the
procedures employed, despite the apparent infringements
of the ordinary rules of reasoning. Leïbnitz, in a hurry to
answer, had himself presented a completely erroneous explanation,
saying that he treated the infinitely small as
_incomparables_, and that he neglected them vis-à-vis finite quantities
_like grains of sand in relation to the sea_, a consideration which would have
completely distorted his analysis, reducing it to being no more than a
simple approximation calculation, which, in this respect, would be radically
vicious, since it would be impossible to predict , in general thesis, to
what extent successive operations can magnify these errors
first, the increase of which could even obviously thus become
arbitrary. Leibnitz had therefore only glimpsed in an extremely
confused manner the true rational foundations of the analysis which he had
created. Its first successors limited themselves at first to verifying
its accuracy by the conformity of its results, in certain
particular uses , with those provided by ordinary algebra or the
geometry of the ancients, by reproducing, as much as they could,
according to the old methods, the solutions of some problems,
once they had been obtained by the new method, the only one
capable originally of making them discover. When this great
question has been considered in a more general way, the geometers,
instead of directly approaching the difficulty, preferred to elude it in
a way, as did Euler and d'Alembert, for example, by
demonstrating abstractly the necessary and constant conformity
of Leïbnitz's conception, considered in all its uses whatsoever, with
other fundamental conceptions of transcendent analysis, that of
Newton above all, the accuracy of which was safe from all objection. One
such general check is probably strictly sufficient to
dispel any uncertainty on the legitimate use of the analysis
Leibnitz. But the infinitesimal method is so important,
it still presents, in almost all the applications, such an
effective superiority over the other general conceptions
successively proposed, that there would be truly imperfection in
the philosophical character of science not to be able to justify it in
itself, and to found it logically on considerations of another
order, which we would then cease to use effectively. It was therefore
of real importance to establish directly and in a general way
the necessary rationality of the infinitesimal method. After various
more or less imperfect attempts to achieve this, the
philosophical works of Lagrange having strongly postponed, towards the end of the
last century, the attention of geometers to the general theory of
infinitesimal analysis, a highly recommendable geometer, Carnot,
finally presented the true direct logical explanation of
Leïbnitz's method , showing it to be based on the principle of compensation.
necessary errors, which is probably, in fact, the
precise and luminous manifestation of what Leïbnitz had vaguely and
confusedly glimpsed when he conceived the rational bases of his analysis.
Carnot thus rendered an essential service to science [13],
the importance of which seems to me not yet sufficiently appreciated,
although, as we will see at the end of this lesson, all this
The logical scaffolding of the infinitesimal method properly so called is
very likely only susceptible of a provisional existence,
as radically vicious by its nature. I nevertheless believe,
however, that I should consider here, in order to complete this important
exposition, the general reasoning proposed by Carnot, to
directly legitimize Leïbnitz's analysis. Here's what it
basically consists of .
[Note 12: One cannot contemplate, without a deep
interest, the naive enthusiasm of the illustrious Huyghens,
about this admirable creation, although his advanced age does
not allow him to make any important use himself,
and that without this powerful aid he would have risen to
capital discoveries. _I see with surprise and
admiration_, he wrote in 1692 to the Marquis de L'Hôpital,
_the extent and fruitfulness of this art; whichever way
I turn my view, I see new uses; finally,
I conceive of it an
infinite progress and an infinite speculation._]
[Note 13: See the remarkable work he published
under the title of _Reflexions sur la Métaphysique du calcul infinitesimal_, and in which we find moreover a
clear and useful exposition, though too little depth, of
all the various points of view from which the
general system of the calculus of indirect functions.]
When we establish the differential equation of a phenomenon, we substitute
for the immediate elements of the various quantities considered, other
simpler infinitesimals which differ infinitely little from
them, and this substitution constitutes the principal artifice of the
method of Leïbnitz, which, without it, would offer no real facility
for the formation of the equations. Carnot regards such a hypothesis
as truly producing an error in the equation thus obtained,
and which, for this reason, he calls _imparfaite_; only, it is
clear that this error can only be infinitely small. Now, from another
On the other hand, all the analytical procedures, either of differentiation or
of integration, which are applied to these differential equations to
rise to finite equations by eliminating all the infinitesimals
introduced as auxiliaries, also constantly produce, by their
nature, as well as 'it is easy to see other analogous errors,
so that an exact compensation has been able to take place, and that the
final equations can, according to Carnot's expression, have become
_perfect_. Carnot considers as a certain and invariable symptom of
the effective establishment of this necessary compensation, the
complete elimination of the various infinitely small quantities, which is constantly,
indeed, the final goal of all the operations of
transcendent analysis . For, if we have never committed other infringements of the
general rules of reasoning than those thus required by the very nature
of the infinitesimal method, the infinitely small errors
produced in this way have never been able to generate other than
infinitely small errors. in all the equations, the relations are
necessarily of a rigorous exactitude as soon as they no longer take
place
except between finite quantities, since it can obviously only exist then finite errors, while there is no could
not have arisen . All this general reasoning is based on the notion
infinitesimal quantities, conceived as indefinitely decreasing,
when those from which they derive are considered as fixed.
Thus, to clarify this abstract exhibition with a single example,
let us take up the question of tangents, which is the easiest to analyze
completely. We will look at the equation t = / frac {dy} {dx} obtained above
as affected by an infinitely small error, since it would only be
completely rigorous for the secant. Now, we will complete the
solution by finding, according to the equation of each curve, the relationship
between the differentials of the coordinates. If this equation is, I
suppose, y = ax ^ 2, we will obviously have / [dy = 2axdx + dx ^ 2. /]
In this formula, we will have to neglect the term dx ^ 2 as infinitely
small of the second order. From then on the combination of the two equations
_imperfaites_ / [t = / frac {dy} {dx}, /; dy = 2axdx, /] sufficient to
completely eliminate the infinitesimals, the finite result t = 2ax will
necessarily be rigorous by the effect of the exact compensation of the two
errors committed since it could not, by its nature, be affected
by an infinitely small error, the only one nevertheless that could be,
according to the spirit of the procedures which were followed.
It would be easy to uniformly reproduce the same reasoning in
relation to all the other general applications of
Leïbnitz's analysis .
This ingenious theory is undoubtedly more subtle than solid, when
one seeks to deepen it. But in reality, however, it has no other
radical logical flaw than that of the infinitesimal method itself, of
which it is, it seems to me, the natural development and the
general explanation , so that it must also be adopted. for a long time it
will be considered convenient to employ this method directly.
I now pass to the general exposition of the two other
fundamental conceptions of transcendent analysis, limiting myself for each to
the main idea, the philosophical character of this analysis having
been, moreover, sufficiently determined above, of after conception
of Leïbnitz, to which I had to particularly focus, because it
allows us to grasp it more easily as a whole, and to describe it
more quickly.
Newton successively presented, in several different forms, his
own way of conceiving of transcendent analysis. The one which is
most commonly adopted today, at least among the geometers of the
continent, was designated by Newton, sometimes under the name of _method of the
first and last reasons_, sometimes under that of _method of
limits_, which one uses more frequently.
From this point of view, the general spirit of transcendent analysis
consists in introducing as auxiliaries, instead of quantities
primitive or concurrently with them, to facilitate the establishment
of the equations, the limits of the ratios of the simultaneous increases of
these quantities, or, in other words, the last reasons of these
increases, limits or last reasons which can easily be shown
as having a determined and finite value. A special calculus, which is
the equivalent of the infinitesimal calculus, is then intended to rise from
those equations between these limits to the corresponding equations between the
primitive quantities themselves.
The ability of such an analysis to express more easily
the mathematical laws of phenomena is, in general, that the
calculation bearing, not on the increases themselves of the quantities proposed,
but on the limits of the ratios of these increases, one can
always substitute for each increase any other quantity
simpler to consider, provided that their last reason is the reason
of equality, or , in other words, let the limit of their relation be
unity. It is clear, in fact, that the calculation of the limits can in
no way be affected by this substitution. Starting from this principle,
we find more or less the equivalent of the facilities offered by
Leïbnitz's analysis , which are only then conceived from another point of
view. Thus, the curves will be considered as the limits of a sequence
of rectilinear polygons, movements varied like the limits of a
set of uniform movements more and more near, & c.
Let us want, for example, to determine the direction of the tangent to a
curve; it will be regarded as the limit towards which a
secant would tend , which would revolve around the given point, so that its second
point of intersection would approach the first one indefinitely. By naming
/ Delta y and / Delta x the differences of the coordinates of the two points, we
would have, at each moment, for the trigonometric tangent of the angle
that the secant makes with the abscissa axis, t = / frac {/ Delta y} {/ Delta
x}; from where, by taking the limits, one will deduce, relatively to the tangent
itself, this general formula of transcendent analysis / [t =
L / frac {/ Delta y} {/ Delta x}; [14] /] according to which the calculation of
indirect functions will teach, in each particular case, when
the equation of the curve is given, to deduce the relation between t and x,
eliminating the auxiliary quantities introduced. If, to complete the
solution, we suppose that y = ax ^ 2 is the equation of the proposed curve,
we will obviously have, / [/ Delta y = 2ax / Delta x + (/ Delta x) ^ 2; /] d 'where we
will conclude / [/ frac {/ Delta y} {/ Delta x} = 2ax + / Delta x. /] Now, it is
clear that the limit towards which the second member tends, as
/ Delta x decreases, is 2ax. We will therefore find by this method, t = 2ax,
as we obtained above for the same case, according to the analysis
of Leïbnitz.
[Note 14: I use the characteristic L to designate
the limit.]
Similarly, when we seek the rectification of a curve, we must
substitute for the increase of the arc s, the chord of this increase,
which is obviously with him in such a relation that the limit of
their relation is unity, and then we find, by following moreover the
same course as with the method of Leïbnitz, this general equation of
rectifications / [/ left (L / frac {/ Delta s} {/ Delta x} / right) ^ 2 = 1 +
/ left (L / frac {/ Delta y} {/ Delta x} / right) ^ 2 /] or / [/ left (L / frac {/ Delta
s} {/ Delta x} / right) ^ 2 = 1 + / left (L / frac {/ Delta y} {/ Delta x} / right) ^ 2 +
/ left (L / frac {/ Delta z} {/ Delta x} / right) ^ 2, /] depending on whether the curve is
plane or double curvature. It will now be necessary, for each
particular curve , to pass from this equation to that between the arc and the abscissa,
which depends on the transcendent calculation proper.
We would resume with the same facility, according to the method of limits,
all the other general questions, the solution of which has been indicated
above, according to the infinitesimal method.
This is, in essence, the conception that Newton had formed
for transcendent analysis, or, more exactly, that which Maclaurin
and d'Alembert have presented as the most rational basis for this
analysis, seeking to fix and coordinate Newton's ideas on this
subject.
I must, nevertheless, before proceeding to the exposition of
Lagrange's conception , point out here another distinct form in which Newton
presented this same method, and which deserves to fix
our attention particularly , so much for its ingenious clarity in some cases ,
as having furnished the most appropriate notation for this way
of considering transcendent analysis, and finally, as being still
today the special form of the calculation of indirect functions
commonly adopted by English geometers. I want to talk about the calculation
des _fluxions_ and _fluentes_, based on the general notion of
_vitesses_.
To make it easier to conceive of the parent idea, let us consider
any curve as generated by a point animated by a movement varied
according to any law. The various quantities which the curve can
offer, the abscissa, the ordinate, the arc, the area, etc., will be considered
as simultaneously produced by successive degrees during this
movement. The _speed_ with which each one will have been described will be called
the _fluxion_ of this quantity, which, conversely, would be called
the _fluent_. Consequently, the transcendent analysis will consist, in this
conception, in immediately forming the equations between the fluxions of
quantities proposed to then deduce, by a special calculation, the
equations between the fluents themselves. What I have just stated in
relation to curves can moreover evidently be transported to
any magnitude, considered, with the aid of a suitable image,
as produced by the movement of one another.
It is easy to understand the general and necessary identity of this
method with that of limits, complicated by the foreign idea of
movement. Indeed, taking the case of the curve, if we assume,
as we can obviously always do, that the movement of the
describing point is uniform in a certain direction, for example,
in the direction of the abscissa, then the fluxion of the abscissa will be constant,
like the element of time. For all the other generated quantities,
the motion could only be conceived as uniform for an
infinitely small time . This being said, the speed being generally, according to its
mechanical notion, the ratio of each space to the time used to
cover it, and this time being here proportional to the increase of
the abscissa, it follows that the fluxion of the ordinate, of the arc, of
the area, etc., are really nothing else, by eliminating
the intermediate consideration of time, than the last reasons for the
increases of these various quantities compared to that of the abscissa.
This method of fluxions and fluents is therefore in reality only a
way of representing, according to a mechanical comparison, the method
of the first and last reasons, which alone is reducible in calculation.
It therefore necessarily entails the same general advantages in the
various principal applications of transcendent analysis, without
our needing to specifically note it.
Finally, I consider Lagrange's conception.
It consists, in its admirable simplicity, in representing the
transcendent analysis as a great algebraic artifice, according to which, to
facilitate the establishment of the equations, we introduce, instead of
primitive functions or with them, their _dérivées_ functions,
that is to say, according to Lagrange's definition, the coefficient of the
first term of the increase of each function, ordered according to the
ascending powers of the increase of its variable. The calculation of
indirect functions proper, is always intended, as
in the conceptions of Leïbnitz and Newton, to eliminate these _dérivées_
employed as auxiliaries, in order to deduce from their relations the
corresponding equations between the primitive quantities.
Transcendent analysis is then nothing more than a simple
very considerable extension of ordinary analysis. It was already a long time ago
a procedure familiar to geometers, to introduce into
analytical considerations, instead of the very magnitudes which they had to
study, their various powers, or their logarithms, or their
sines, etc., in order to simplify the equations, and even to obtain them
more easily. Successive _ derivation_ is a general artifice of the
same nature, which presents only much more scope, and
consequently procures for this common end much greater resources
.
But, although one can conceive without doubt _a priori_ that the
auxiliary consideration of these derivatives, _may_ facilitate the establishment of the
equations, it is not easy to explain why this _should_ be
necessarily according to the mode of derivation adopted rather than according to
any other transformation. This is the weak side
of Lagrange's great thought . We have not, in fact, really succeeded so far in
grasping in general in an abstract manner, and without entering into the
other conceptions of transcendent analysis, the precise advantages which
must constantly present, by its nature, this analysis thus conceived,
for the research of the mathematical laws of the phenomena. It is
only possible to ascertain them, by considering each
main question separately , and this verification becomes even tedious, when
one chooses a complicated question.
To briefly indicate how this way of conceiving of
transcendent analysis can effectively adapt to the solution of
mathematical problems , I will limit myself to taking up again from this point of view the
simplest problem of all those examined above, that of
tangents. .
Instead of conceiving the tangent as the prolongation of the
infinitely small element of the curve, according to the notion of Leïbnitz; or as
the limit of the secants, according to Newton's ideas; Lagrange
considers it according to this simple geometric character, analogous to the
definitions of the ancients, of being a straight line such that between it and the
curve it cannot pass, through the point of contact, any other straight line.
Therefore, to determine its direction, it is necessary to seek the
general expression of its distance from the curve, in any direction, in that
of the ordinate, for example, at a second point distinct from the first, and to
have the arbitrary constant relative to the inclination of the
line, which will necessarily enter into this expression, so as to
reduce this distance as much as possible. Now, this distance being
obviously equal to the difference of the two ordinates of the curve and of
the straight line which correspond to the same new abscissa x + h, will be
represented by the formula / [(f '(x) -t) h + qh ^ 2 + rh ^ 3 + / mbox {/ rm
etc.}, /] Where t denotes, as above, the trigonometric tangent
unknown of the angle made with the axis of (x), the line sought, and
f '(x), the function derived from the ordinate f (x). That said, it is easy to
see that by having t so as to cancel the first term of the
preceding formula, we will have made the interval of the two lines as
small as possible, so much so that any other line for which t
n ' would not have the value thus determined, would necessarily deviate
further from the proposed curve. We therefore have, for the direction of the
tangent sought, the general expression t = f '(x); result exactly
equivalent to those provided by the infinitesimal method, and the
limit method. It will remain now, in every curve
particular, to find f '(x), which is a pure question of analysis,
completely identical with those which then prescribe the other
methods.
After having sufficiently considered as a whole the principal
general conceptions successively produced so far for the
transcendent analysis , I must not stop at the examination of some other
theories proposed, such as the _calcul of the evaporators_ of Euler, which
are really only more or less important modifications, and
moreover unusual, of the preceding methods. In
order to complete this set of considerations, it remains for me to establish the
comparison and the appreciation of these three fundamental methods.
must first ascertain in general, their
perfect and necessary conformity .
It is first evident, from what precedes, that considering these three
methods as to their effective destination, independently of the
preliminary ideas , they all consist of the same
general logical artifice , which I characterized in the fourth lesson, namely:
the introduction of a certain system of auxiliary quantities,
uniformly correlative to those which are the proper object of the
question, and which one expressly substitutes for them to facilitate
the analytical expression of the mathematical laws of phenomena,
although they must ultimately be eliminated, using a calculation
special. This is what determined me to regularly define the
transcendent analysis _the calculus of indirect functions_, in order to mark its
true philosophical character, by ruling out any discussion on
the most suitable way of conceiving and applying it. The
general effect of this analysis, whatever the method employed, is therefore
to bring each mathematical question much more promptly
into the domain of _calcul_, and thus to considerably reduce the
capital difficulty which the passage from the concrete to
the concrete usually presents . abstract. Whatever one does, one cannot hope that the calculus will
never take hold of every question of natural, geometric, or
mechanical, or thermological, etc., immediately upon its birth, which
would obviously be contradictory. In any
problem there will constantly be some preliminary work to be done without the calculation
being of any help, and which cannot, by its nature, be
subject to abstract and invariable rules; it is that which has as its
own object the establishment of the _equations_, which are the
essential starting point of all analytical research. But this
preliminary elaboration was singularly simplified by the creation of
the transcendent analysis, which thus hastened the era when the solution included
the uniform and precise application of general procedures and
abstract; by reducing, in each case, this special work to the
search for equations between the auxiliary quantities, from which the calculation
then leads to the equations directly relating to the quantities
proposed, which, before this admirable conception, had to be established
immediately. Let these indirect equations be
_differential_ equations , according to Leïbnitz's thought; or equations _at the
limits_, according to Newton's ideas; or finally
_developed_ equations , according to Lagrange's theory; the general procedure is
obviously always the same.
But the coincidence of these three main methods is not limited to
the common effect they produce; it exists, moreover, in the
very way to get it. In fact, not only do all three
consider, in place of the primitive quantities, certain
auxiliary quantities ; moreover, the quantities thus introduced in the alternative
are exactly identical in the three methods, which differ,
consequently, only in the manner of considering them. This can
easily be seen, taking
any of the three conceptions as a general term of comparison , especially that of Lagrange, the most
suitable to serve as a type, as being the most free from
foreign considerations . Is it not obvious, by the sole definition of _functions derived_
, that they are nothing other than what Leïbnitz calls the
_differential coefficients_, or the ratios of the differential of
each function to that of the corresponding variable, since, in
determining the first differential, we must, by the very nature of
the infinitesimal method, limit ourselves to taking the sole term of
the increase of the function which contains the first power of
the infinitely small increase in the variable? Likewise, the
derived function is not also by its nature the necessary _limit_ towards
which the relation between the increase of the primitive function
and that of its variable tends , as the latter decreases indefinitely,
since it expresses obviously what becomes of this relation, assuming
null the increase of the variable. What we denote by / frac {dy} {dx}
in Leïbnitz's method, what we should denote by L / frac {/ Delta
y} {/ Delta x} in Newton's method, and what Lagrange indicated by
f '(x), is always the same function, considered from three
different points of view; the considerations of Leïbnitz and Newton, consisting
properly in making known two necessary general properties of the
derivative function. Transcendent analysis, examined in the abstract, and
in principle, is therefore always the same, whatever the
conception one adopts: the methods of calculating indirect functions
are necessarily identical in these various methods, which,
likewise, must, for any application,
constantly lead to strictly conforming results.
If now we seek to appreciate the relative value of these three
equivalent conceptions, we will find in each of the advantages and
disadvantages which are specific to it, and which still prevent the
surveyors from sticking strictly to one of them, considered
final.
The conception of Leïbnitz presents, incontestably, in the set
of applications, a very pronounced superiority, leading in a
much more rapid manner, and with much less
intellectual effort , to the formation of the equations between the quantities
auxiliaries. It is to its use that we owe the high perfection which
all the general theories of geometry and
mechanics have at last acquired . Whatever the various speculative opinions of the
geometers on the infinitesimal method, considered abstractly, all
tacitly agree to use it in preference, as soon as they have to
deal with a new question, so as not to complicate the difficulty
necessary by this a purely artificial obstacle, coming from a
misplaced obstinacy in wanting to follow a less hasty course.
Lagrange himself, after having reconstructed
transcendent analysis on new bases , returned, with this high frankness which
suited his genius so well, a dazzling and decisive homage to the
characteristic properties of Leïbnitz's conception, following it
exclusively in the entire system of _mechanics analytic_. One
such exemption us, about this, any other consideration.
But when one considers in itself, and from the logical point
of view, Leïbnitz's conception, one cannot help recognizing with
Lagrange that it is radically vicious, in that, according to its
own expressions, the notion of the infinitely small, is a _ false
idea_, which it is indeed impossible to represent clearly,
although one sometimes deludes oneself in this respect. Analysis
transcendent, to read, this, in my eyes, this great
philosophical imperfection of being still essentially based e
on these metaphysical principles, the human mind has had so much trouble
to clear all its positive theories. In this respect, we can say
that the infinitesimal method really bears the characteristic imprint
of the time of its foundation, and of the genius proper to its founder. One
can well, it is true, by the ingenious idea of the compensation of the
errors, to explain in a general way, as we did
it above, the necessary exactitude of the general processes which compose
the infinitesimal method. . But isn't that alone a downside
radical, than to be obliged to distinguish, in mathematics, two classes
of reasoning, those which are perfectly rigorous, and those in
which we deliberately commit errors which will have to be compensated for
later? A conception which leads to such strange consequences is,
no doubt, rationally, very unsatisfactory.
It would obviously be to elude the difficulty without solving it, to say,
as has sometimes been done, that it is possible, with respect to each
question, to bring the infinitesimal method properly so called
into that of limits, whose character logic is beyond reproach.
Moreover, such a transformation almost entirely removes
Leibnitz's conception of the essential advantages which recommend it so
eminently, as regards the facility and rapidity of
intellectual operations .
Finally, even if we had no regard for the important considerations which
precede, the infinitesimal method would nevertheless
obviously present, by its nature, this capital defect of breaking the unity of
abstract mathematics, by creating a transcendent calculation based on of
principles so different from those which form the basis for the analysis
ordinary. This division of analysis into two almost independent worlds
tends to prevent the formation of truly
general analytical conceptions . To fully appreciate the consequences, it would be necessary to
to defer, by thought, to the state in which science found itself,
before Lagrange had established between these two great sections a
general and definitive harmony.
Turning to Newton's conception, it is obvious that, by its nature,
it is immune from the fundamental logical objections that
Leïbnitz's method provokes. The notion of _limits_ is, in fact,
remarkable for its clarity and accuracy. In
transcendent analysis presented in this way, the equations are regarded
as correct from the outset, and the general rules of reasoning
are as consistently observed as in ordinary analysis. But,
on the other hand, it is far from offering,
problems, as powerful resources as the infinitesimal method.
This obligation which it imposes never to consider the
increases of magnitudes separately and in themselves, nor only
in their relations, but only within the limits of these relations,
considerably slows down the progress of the intelligence for the formation
of the auxiliary equations. We can even say that it greatly hinders
purely analytical transformations. Also the transcendent calculus,
considered separately from its applications, is far from offering in
this method the extent and the generality which the
conception of Leïbnitz gave it. It is very painfully, for example, that we
manages to extend Newton's theory to functions of several
independent variables. However that may be, it is above all in relation
to the applications that the relative inferiority of this theory is
marked.
I must not neglect on this subject to observe that several
geometers of the continent, by adopting, as more rational, the method
of Newton, to serve as a basis for the transcendent analysis,
partly disguised this inferiority, by a serious inconsistency , which consists in
applying to this method the notation imagined by Leïbnitz for the
infinitesimal method, and which is really specific only to it. By
designating by / frac {dy} {dx} what, rationally, it would take, in
limit theory, denote by L / frac {/ Delta y} {/ Delta x}, and by extending
this displacement of signs to all other analytical notions, we
no doubt propose to combine the special advantages of the two methods;
but in reality one only succeeds in establishing between them a
vicious confusion , the habit of which tends to prevent the formation of clear
and exact ideas of one or the other. It would undoubtedly be strange,
considering this usage in itself, if, by the only means of signs, one
could effect a true combination between two general theories
so distinct.
Finally, the method of limits also presents, although to a lesser degree,
the major drawback that I pointed out above, in the
infinitesimal method , of establishing a total separation between
ordinary analysis and transcendent analysis. For the idea of _limits_, although
clear and rigorous, is nonetheless, by itself, as Lagrange
observed, a foreign idea, on which analytical theories
should not be dependent.
This perfect unity of analysis, this purely abstract character of its
fundamental notions, is to be found in the highest degree in
Lagrange's conception , and only to be found there. It is, for this reason, the
most rational and the most philosophical of all. Carefully spreading
Considering all heterogeneous considerations, Lagrange reduced
transcendent analysis to its own true character, that of offering a
very extensive class of analytical transformations, with the help of which
the expression of the conditions of the various
problems is singularly facilitated . At the same time, this analysis has necessarily presented itself
as a simple extension of ordinary analysis; it was no more
than a superior algebra. All the various, hitherto
inconsistent parts of abstract mathematics could be conceived from
that moment on as forming a single system.
Unfortunately, a gifted design, regardless of the scoring though
simple and so lucid which corresponds to it, of properties so
fundamental, and which is, without doubt, destined to become the
definitive theory of the transcendent analysis, because of its high
philosophical superiority on all the other methods proposed, present in
its state current, too many difficulties, as for the applications, when
compared to the conception of Newton, and especially that of Leïbnitz,
to be able to be still exclusively adopted. Lagrange himself only
succeeded very painfully in rediscovering, according to his method, the
principal results already obtained by the infinitesimal method for the
solution of general questions of geometry and mechanics; we can
to judge by this how many obstacles one would find to treat, in the same
way, questions really new and of some importance. It
is true that Lagrange, on several occasions, has shown that
difficulties, even artificial ones, determine, in men of genius,
superior efforts, capable of leading to more
extensive results . It is in this way that by trying to adapt his method to the study of the
curvature of lines, which seemed so little to be able to include
the application of it, he rose to this beautiful theory of contacts, which has
perfected so much this important part of geometry. But, despite
these happy exceptions, Lagrange's conception is no less
hitherto remained, as a whole, essentially unsuitable for
applications.
The final result of the general comparison which I have just sketched,
and which would require further development, is therefore, as I
had advanced at the beginning of this lesson, that, in order to really know
transcendent analysis, one must not only to consider it, in
principle, according to the three distinct fundamental conceptions,
produced by Leïbnitz, by Newton, and by Lagrange, but, moreover, to
become accustomed to follow almost indifferently according to these three
principal methods , and especially according to the two extremes, the solution of all
the important questions,
itself, or its applications. It is a walk that I cannot
recommend highly enough to all those who wish to
philosophically judge this admirable creation of the human mind, as well as to
those who primarily want to learn how to use
this powerful instrument successfully and with ease. In all other parts
of mathematical science, the consideration of various methods for
a single class of questions can be useful, even regardless of
its historical interest; but it is not
essential: here, on the contrary, it is strictly necessary.
Having determined with precision, in this lesson, the character
philosophy of the calculus of indirect functions, according to the
principal fundamental conceptions of which it is susceptible, it
remains for me now to consider, in the following lesson, the
rational division and the general composition of this calculus.
SEVENTH LESSON.
SUMMARY. General table of the calculation of indirect functions.
As a result of the considerations exposed in the previous lesson, we
can see that the calculation of indirect functions necessarily divides
into two parts, or, to put it better, decomposes into two
quite distinct calculations , although, by their nature, intimately related;
depending on whether one proposes to find the relations between the quantities
auxiliaries, the introduction of which constitutes the general spirit of this
calculation, according to the relations between the
corresponding primitive quantities ; or that we seek, in the opposite direction, to discover these
direct equations from the indirect equations established
immediately. Such is, in fact, the double object that we
constantly have in view in transcendent analysis.
These two calculations have received different names, depending on the point of view from
which the whole of this analysis has been considered. The
infinitesimal method properly so called being so far the most used, for the
reasons which I have discussed, almost all the geometers of the continent
usually employ, to designate these two calculations,
The denominations of _differential calculus_ and of _ integral calculus_, established
by Leibnitz, and which are, in fact, very rational consequences
of his conception. Newton, after his method, named the first the
_calcul des fluxions_, and the second the _calcul des fluentes_,
expressions commonly adopted in England. Finally, following the
eminently philosophical theory founded by Lagrange, one would call
the _calculus of derivative functions_ and the other the _calculus of
primitive functions_. I will continue to use the terms of
Leïbnitz, as more appropriate, in our language, to the formation of the
secondary expressions, although I must, according to the explanations
contained in the previous lesson, to use concurrently all the
various conceptions, while approaching, as much as possible, that
of Lagrange.
Differential calculus is obviously the rational basis of
integral calculus . For we only know and cannot know how to integrate immediately
the differential expressions produced by the differentiation of the
various simple functions which constitute the general elements of our
analysis. The art of integration then consists essentially in
reducing, as much as possible, all the other cases to
ultimately depend only on this small number of fundamental integrations.
Considering the whole of the transcendent analysis, as I have it
characterized in the previous lesson, we do not see at first what
the proper utility of the differential calculus can be, independently of this
necessary relation with the integral calculus, which seems to have to be, by
itself, the only one directly indispensable. In fact, the elimination of
infinitesimals or of derivatives, introduced as auxiliaries to
facilitate the establishment of the equations, constituting, according to what
we have seen, the final and invariable object of the calculation of
indirect functions ; it is natural to think that the calculation which teaches to
deduce from the equations between these auxiliary quantities, those which take
place between the primitive quantities themselves, must strictly
be sufficient for the general needs of transcendent analysis, without our
perceiving, at first glance, what special and constant part
the solution of the opposite question can have in such an analysis.
It would be wrong if, according to ordinary usage, in order to explain
the direct and necessary influence proper to differential calculus, it would be
assigned the purpose of forming the differential equations,
from which the integral calculus then leads to finite equations. For
the primitive formation of differential equations is not, and cannot
be, strictly speaking, the object of any calculation, since it
constitutes, on the contrary, by its nature, the indispensable starting point.
of any calculation. How, in particular, could
differential calculus which, by itself, be reduced to teaching the means of
_differentiating_ the various equations, be a
general process for establishing them? What, in any application of
transcendent analysis , indeed facilitates the formation of equations, is the
infinitesimal _method_, and not the infinitesimal _calcul_, which is
perfectly distinct from it, although being the indispensable complement. One
such consideration so would misrepresent the destination
special characteristic differential calculus in the general system
of transcendental analysis.
But it would be, nevertheless, to conceive very imperfectly the true
proper importance of this first branch of the calculus of
indirect functions , to see in it only a simple preliminary work,
having no other general and essential object than to prepare for the
integral calculus. indispensable foundations. As ideas are
usually confused in this respect, I believe it is my duty to explain
briefly here this important relation, as I understand it, and to
show that, in every application whatever of
transcendent analysis , a first direct and necessary part is constantly
assigned to the Differential calculation.
By forming the differential equations of any phenomenon, he
It is very rare that one confines oneself to introducing differentially the only
quantities whose relations one seeks. To impose this condition
would be to reduce unnecessarily the resources which
transcendent analysis presents for the expression of the mathematical laws of phenomena.
Most often we also introduce by their differentials, in these
first equations, other magnitudes, the relation of which is already
known or supposed to be, and without the consideration of which it would
often be impossible to establish the equations. Thus, for
example, in the general problem of rectification of curves,
the differential equation, / [ds ^ 2 = dy ^ 2 + dx ^ 2, / mbox {/ rm or} ds ^ 2 =
dx ^ 2 + dy ^ 2 + dz ^ 2, /] is not only established between the
sought function s and the independent variable x to which we want to
relate it; but we have introduced at the same time, as
indispensable intermediaries , the differentials of one or two other functions y and
z, which are among the data of the problem; it would not have been
possible to immediately form the equation between ds and dx, which would
moreover be specific to each curve considered. The same is true
for most questions. Now, in these cases, it is obvious that
the differential equation is not immediately specific to
integration. It is necessary, first, that the differentials of
supposedly known functions, which have been used as intermediaries,
are entirely eliminated, so that the equations are
established between the differentials of only the functions sought and
those of the truly independent variables, after which the question
effectively depends only on the integral calculus. However, this
preparatory elimination of certain differentials, in order to reduce the
infinitesimals to the smallest possible number, is simply the responsibility
of the differential calculus. For it must be done, obviously, by
determining, from the equations between the supposedly known functions
taken as intermediaries, the relations of their differentials, this
which is only a question of differentiation. Thus, for example, in
the case of rectifications, it will first be necessary to calculate dy or dy and dz, by
differentiating the equation or the equations of each curve proposed; and,
according to these expressions, the general differential formula stated
above will contain only ds and dx; at this point,
the elimination of infinitesimals can no longer be completed except by the
integral calculus.
Such is therefore the general office necessarily proper to
differential calculus in the total solution of the questions which require the use
of transcendent analysis: to prepare, as far as possible, the elimination
of infinitesimals, that is to say to reduce,
primitive differential equations to contain only the
differentials of the truly independent variables and those of the
functions sought, by eliminating, by differentiation, the
differentials of all the other known functions which may have been
taken as intermediaries during the formation of the
differential equations of the problem.
For certain questions, which, although in small number, are nonetheless
, as we shall see later, of very great importance,
the quantities sought are even found to enter directly, and not by
their differentials, into the primitive differential equations,
which then contain differentially only the various
known functions , employed as intermediaries according to the
preceding explanation . These cases are, of all, the most favorable, because, it is
obvious that the differential calculus is then entirely sufficient for
the complete elimination of infinitesimals, without the question being able to
give rise to any integration. This is what happens, for example, in
the problem of tangents, in geometry; in that of speeds, in
mechanics, etc.
Finally, several other questions, the number of which is also very small,
but of no less importance, present a second case
of exception, which is, by its nature, exactly the reverse of the preceding one.
These are those in which the differential equations are found to be
immediately suitable for integration, because they contain,
from their first formation, only the infinitesimals relating to the
sought functions or to the truly independent variables, without
having been obliged to d 'differentially introduce other functions
as intermediaries. If, in these new cases, we have effectively used
these latter functions, as, by hypothesis, they will enter
directly and not by their differentials, ordinary algebra
will suffice to eliminate them, and reduce the question to depend only
on the integral calculus. . The differential calculus will therefore have no part
special to the complete solution of the problem, which will be entirely the
spring of the integral calculation. The general question of quadratures
offers an important example, because the differential equation being then,
dA = ydx, will immediately become suitable for integration as soon as one
eliminates, from the equation of the proposed curve, the function
intermediary y, which does not enter there differentially: the same
circumstance takes place for the problem of cubatures, and for some
others as essential.
As a general result of the preceding considerations, it is therefore necessary to
divide into three classes the mathematical questions which require
the use of transcendent analysis: the first class comprises the
problems which can be entirely solved by means of the
differential calculus alone , without any need for the integral calculus; the second,
those which are, on the contrary, entirely within the competence of the integral calculus,
without the differential calculus having any part in their solution; finally,
in the third and most extensive, which constitutes the normal case, the
other two being only of exception, the two calculations successively have
a distinct part and necessary for the complete solution of the problem, the
differential calculus subjecting to the
primitive differential equations , an essential preparation for the application of
integral calculus . Such are exactly the general relations of these two
calculations, of which we commonly form ideas that are too imprecise.
Let us now take a general look at the rational composition
of each of them, starting, of course, with the
differential calculus.
In the exposition of transcendent analysis, we are accustomed to mixing with
the purely analytical part, which is reduced to the abstract treatise on
differentiation and integration, the study of its various
main applications, especially those which concern geometry.
This confusion of ideas, which is a consequence of the effective mode according to
which science has developed, presents, under the
dogmatic point of view, serious disadvantages in that
suitably, either analysis or geometry. Having to consider here
the most rational coordination possible, I will understand, in the
following table, only the calculation of indirect functions properly speaking,
reserving, for the portion of this volume relating to the philosophical study
of concrete mathematics, the examination general of its major
geometric and mechanical applications [15].
[Note 15: I have long since established, in my
ordinary teaching of transcendent analysis, the order
which I am going to present. A new professor of
transcendent analysis at École Polytechnique, with whom I am
delighted to have met, M. Mathieu adopted,
during this year, an essentially similar course.]
The fundamental division of pure differential calculus, or of the
general treatise on differentiation, consists in distinguishing two cases, according to
whether the analytical functions to be differentiated are
_explicit_ or _implicit_ ; hence two parts usually designated
by the names of differentiation _of formulas_ and differentiation _of
equations_. It is easy to conceive _a priori_ the importance of this
classification. Indeed, such a distinction would be illusory if
ordinary analysis were perfect, that is to say, if one knew how to
solve all the equations algebraically; because then it would be
possible to _explicit_ any _implicit_ function; and, by
differentiating it only in this state, the second part of the
differential calculus would immediately enter into the first, without giving rise
to any new difficulty. But the algebraic resolution of the
equations being, as we have seen, still almost in childhood,
and ignored until now for the greatest number of cases, we
understand that it must be quite otherwise; since it is therefore,
strictly speaking, a question of differentiating a function without knowing it, although
it is determined. The differentiation of implicit functions
is therefore, by its nature, a question truly distinct from that
that present the explicit functions, and necessarily more
complicated. Thus it is obviously by the differentiation of the formulas
that we must begin, and we then succeed in generally reducing to this
first case the differentiation of the equations, by certain
invariable analytical considerations, which I should not mention
here.
These two general cases of differentiation are still distinct in
another respect equally necessary, and too important for me to
neglect to mention it. The relation obtained between the differentials
is constantly more indirect, with respect to that of
finite quantities , in the differentiation of implicit functions than in that of
explicit functions. We know, in fact, from the considerations
presented by Lagrange on the general formation of
differential equations , that, on the one hand, the same primitive equation can give
rise to a greater or lesser number of equations derived from
very -various, although, at bottom, equivalent, according to those of the
arbitrary constants which one eliminates, which does not take place in the
differentiation of the explicit formulas; and that, on the other hand, the
infinite system of different primitive equations which correspond to the
same derived equation, presents an analytical variety much
deeper than that of the various functions susceptible of the same
explicit differential, and which are distinguished from each other
only by a constant term. Implicit functions must therefore be
considered as being really even more modified by
differentiation than explicit functions. We will find
this consideration later in relation to the integral calculus, where it
acquires a preponderant importance.
Each of the two fundamental parts of differential calculus is itself
subdivided into two very distinct theories, according to whether it
is a question of differentiating between functions with a single variable, or
functions with several independent variables. This second case is, by its
nature, quite distinct from the first, and obviously presents more
complication, even considering only the explicit functions, and all the
more so for the implicit functions. Moreover, one is
generally deduced from the other, with the aid of a very
simple invariable principle , which consists in looking at the total differential of a function
by virtue of the simultaneous increases of the various
independent variables which it contains. , as the sum of the
partial differentials which the separate increase of each variable
successively would produce , if all the others were constant. We must,
moreover, carefully note a new notion on this subject.
introduced into the system of transcendent analysis by the distinction
between functions with a single variable and with several: it is the
consideration of these various special derived functions, relating to
each variable separately, and the number of which increases in addition in addition
as the order of the derivation increases, and also as the
variables are more multiplied. It follows that the
differential relations specific to the functions of several variables are, by
their nature, and much more indirect, and above all much more
indeterminate than those relating to the functions of a single variable.
This is mainly sensitive for implicit functions where, at
instead of the simple arbitrary constants that elimination makes
disappear when we form the differential equations specific to the
functions of a single variable, it is arbitrary functions of the
proposed variables which are eliminated, from which must result,
during the integrations, special difficulties.
Finally, to complete this summary table of the various
essential parts of the differential calculus proper, I must add that,
in the differentiation of implicit functions, either with a single
variable or with several, it is still necessary to distinguish the case where it
is a question of to differentiate at the same time various functions of this kind, mixed
in some primitive equations, that where all these functions
are separated.
The functions are obviously, in fact, even more implicit in the
first case than in the second, if we consider that the same
imperfection of ordinary analysis, which prevents converting any
implicit function into an equivalent explicit function, does not allow nor
to separate the functions which enter simultaneously into
any system of equations. It is then a question of differentiating,
not only without knowing how to solve the primitive equations, but even
without being able to carry out suitable eliminations between them, which
constitutes a new difficulty.
Such then are the natural sequence and the rational distribution of the
various principal theories of which the general treatise on
differentiation is composed . We see that, the differentiation of
implicit functions being deduced from that of explicit functions by a single
constant principle, and the differentiation of functions with several
variables being reduced, by another fixed principle, to that of functions
with a single variable, the whole Differential calculus turns out to be based, in the
final analysis, on the differentiation of explicit functions with a
single variable, the only one that never directly executes. Now, it is
easy to imagine that this first theory, the necessary basis of the system
integer, consists simply in the differentiation of the ten
simple functions , which are the uniform elements of all our
analytical combinations , and whose table I have presented (4th lesson, page 173). For
the differentiation of the compound functions is obviously deduced, in an
immediate and necessary way, from that of the simple functions which
constitute them. It is therefore to the knowledge of these ten
fundamental differentials , and to that of the two general principles,
mentioned above , which bring together all the other possible cases, that the
whole treaty of differentiation is reduced, strictly speaking. We see, by
the combination of these various considerations, how much is at the same time
simple and perfect the entire system of differential calculus proper
. It certainly constitutes, from a logical point of view, the
most interesting spectacle that mathematical analysis can present to our
intelligence.
The general picture that I have just sketched out would
nevertheless offer an essential gap, if I did not indicate here clearly
a last theory, which forms, by its nature, the
indispensable complement of the treatise on differentiation. It is that which has for
object the constant transformation of the derived functions, as a result
of the determined changes of independent variables, from which results the
possibility of relating all the formulas to new variables.
general differentials originally established for others. This
question is now solved in the most complete and
simplest way, like all those of which differential calculus is composed.
We can easily see the general importance that it must have in
any applications of transcendent analysis, of which it can be
considered as increasing the fundamental resources, by making it possible
to choose, in order to form the
differential equations more easily , the system of independent variables which will appear the
most advantageous, although it should not be maintained later. It is
, for example, that most major issues
geometry are much more easily solved by relating lines
and surfaces to rectilinear coordinates, and that one can nevertheless
be led to apply them to forms expressed analytically,
using _polar_ coordinates, or in any other way. We can
then begin the differential solution of the problem by
always employing the rectilinear system, but only as an intermediary,
according to which, by the general theory which we have in view here, we
will pass to the definitive system, which it would sometimes have been. impossible to
consider directly.
In the rational classification that I have just exposed for
the whole of the differential calculus,
point out a serious omission, since I have not sub-divided each of the
four essential parts according to another general consideration,
which at first seems very important in itself, that of the more
or less elevated order of the differentiation. But it is easy to understand that
this distinction has no real influence in the
differential calculus , in that it does not give rise to any new difficulty.
Indeed, if the differential calculus was not rigorously complete,
that is to say, if one did not know how to differentiate indiscriminately any
function whatever, the differentiation to the second order, or to a
higher order , of each determined function, could generate
special difficulties. But the perfect universality of
differential calculus obviously gives the assurance of being able to differentiate
all known analytical functions at any order, the question
being constantly reduced to a differentiation at the first order,
successively redoubled. Thus, the consideration of the various orders of
differentials may well give rise to new more
or less important remarks , especially as regards the formation of
differential equations, and the successive partial derivatives of
functions with several variables. But it cannot, of course,
constitute any new general problem in the treaty of
differentiation. We will see later that this distinction, which
has, so to speak, no importance in the differential calculus,
acquires, on the contrary, a very great one in the integral calculus, by
virtue of the extreme imperfection of this last calculation.
Finally, although I believed, as a general thesis, that I should not
at this moment consider the various principal applications of
differential calculus , it is nevertheless advisable to make an exception for those
which consist in the solution of purely analytical questions, which
must, therefore, be made. effect, to be rationally placed following the treaty of
differentiation proper, because of the obvious homogeneity
considerations. These questions can be reduced to three
essentials: 1 ° the development in series of functions with only one or
several variables, or, more generally, the transformation of
functions, which constitutes the most beautiful and the most important application
of differential calculus to the general analysis, and which includes, in addition to the
fundamental series discovered by Taylor, the remarkable series
found by Maclaurin, by Jean Bernouilli, by Lagrange, etc .; 2 ° the
general theory of maximum and minimum values for
any functions with one or more variables, one of the most
interesting problems that analysis can present, whatever
elementary that it has become today, and to the complete solution of
which the differential calculus applies very naturally; 3 ° finally,
the general determination of the true value of the functions which
present themselves under an indeterminate appearance for certain assumptions
made on the values of the corresponding variables, which is
the least extensive and least important problem of the three, although it
deserves to 'be noted here. The first question is, without a doubt, the
main one in all respects: it is also the most likely
to acquire a new extension in the future, especially by designing,
in a broader way than has been done up to now. 'here, the use of calculation
differential for the transformation of functions, about which
Lagrange left some valuable indications, which have not yet been
generalized or followed.
I very much regret to be obliged, by the necessary limits of this
work, to confine myself to such insufficient summary considerations
on all the various subjects which I have just reviewed, and which
would, by their nature, involve much more extensive development.
extended, while still continuing to stay within the generalities
which are the proper subject of this course. I now pass to the
equally rapid exposition of the systematic table of the integral calculus properly
said, that is to say, of the abstract treaty of integration.
The fundamental division of integral calculus is based on the same
principle as that set out above for differential calculus,
distinguishing between the integration of explicit differential formulas, and
the integration of implicit differentials, or
differential equations . The separation of these two cases is even much
deeper in relation to integration, than in the simple matter of
differentiation. In differential calculus, in fact, this
distinction rests, as we have seen, only on the extreme
imperfection of ordinary analysis. But, on the contrary, it is easy to
see that, even if all the equations were solved
algebraically, the differential equations would nonetheless constitute
a case of integration quite distinct from that presented by
the explicit differential formulas. Because, by limiting oneself, for
example, to the first order and to a unique function y of a single
variable x, for simplicity, if one supposes solved, with
respect to / fracdy {dx}, any differential equation between x,
y, and / frac {dy} {dx}, the expression of the derived function
then generally containing the primitive function itself which is
the object of the research, the question of integration would not have not at all
changed in nature, and the solution would have really made no other
progress than to have brought the differential equation, proposed to
be only of the first degree relative to the derivative function, which
is, in itself, of little of importance. The differential would therefore be no
less determined in a manner about as _implicit_ as
before, with regard to integration, which would continue to
present essentially the same characteristic difficulty. The
algebraic resolution of the equations could only include the case
we are considering in the simple integration of
explicit differentials , on the very specific occasions when the equation
proposed differential would not contain the primitive function
itself, which would allow, consequently, by solving it, to
find / frac {dy} {dx} as a function of x only, and thus to reduce the
question to quadratures.
The consideration which I have just indicated for the simplest
differential equations would obviously be even more
important for those of higher orders or which would
simultaneously contain various functions of several independent variables.
Thus, the integration of the differentials which are determined only
implicitly constitutes by its nature, and, without any regard to the state
of the algebra, a case entirely distinct from that relating to
differentials explicitly expressed as a function of the
independent variables . The integration of differential equations is therefore
necessarily more complicated than that of explicit differentials,
by the elaboration of which the integral calculus originated, and on
which we then tried to make
the others depend as much as possible . All the various analytical procedures proposed so far to
integrate the differential equations, either the separation of
variables, or the method of multipliers, etc., in fact
aim to reduce these integrations to those of differential formulas,
the only one which, by its nature, can be undertaken directly.
Unfortunately, however imperfect this
necessary basis of all integral calculus may be, the art of reducing the integration
of differential equations to it is still much less advanced.
Each of these two fundamental branches of integral calculus is then
sub-divided into two others, as in differential calculus,
and by exactly analogous reasons (which I will therefore dispense
with reproducing), depending on whether we consider functions with
a single variable or functions with several independent variables.
I will only observe that this distinction is, like the
preceding one, even more important for integration than for
differentiation. This is especially remarkable in relation to
differential equations. Indeed, those which relate to
several independent variables can obviously present this
characteristic difficulty, and of a much higher order, that the
sought function is defined differentially by a simple
relation between its various special derivatives relating to the various
variables taken separately. From this results the most difficult branch,
and also the most extensive of the integral calculus, what is
usually called the _partial difference integral calculus_, created by
d'Alembert, and in which, according to Lagrange's correct assessment,
geometers should really have seen a new calculation, the
philosophical character of which is not sufficiently accurately judged. A
very salient difference between this case and that of equations with a single
independent variable consists, as I have observed above, in the
arbitrary functions which replace the simple arbitrary constants
to give the corresponding integrals all the
suitable generality .
I hardly need to say that this higher branch of
transcendent analysis is still entirely in infancy, since, only
in the simplest case, that of a first order equation between
the partial derivatives of a only two-variable function
independent, we do not even know until now completely reduce
integration to that of ordinary differential equations.
The integration relating to the functions of several variables is much
more advanced, in the case, infinitely simpler, in truth, where it
is only a question of explicit differential formulas. We then know in
fact, when these formulas fulfill the suitable
integrability conditions , constantly reduce their integration to quadratures.
A new general distinction, applicable, as a sub-division, to
the integration of explicit or implicit differentials, to a single
variable or to several, is drawn from the more or less elevated order of
differentiations, which does not give rise to any special question in the
differential calculus, as we have noticed.
With regard to explicit differentials, either with one variable or with
several, the need to distinguish their various orders is due only to
the extreme imperfection of the integral calculus. In fact, if one knew how to
constantly integrate any first-order differential formula,
the integration of a second-order formula or any other would
obviously not constitute a new question, since by
integrating it first to the first order, we would arrive at the
differential expression of the immediately preceding order, hence, by a sequence
suitable for analogous integrations, one would be certain to
finally go back to the primitive function, the proper object of such work. But
the little knowledge that we have on
first integrations means that this is not the case, and that the more or less
high order of the differentials generates new difficulties. For, having
differential formulas of any order greater than the first,
it may happen that we know how to integrate them a first time or several
times in a row, and that, nevertheless, we cannot thus go back to the
primitive functions, if these works preliminaries have produced, for the
differentials of a lower order, expressions whose
integrals are not known. This circumstance must present itself
all the more frequently, the number of known integrals being still
very small, as these successive integrals are generally, as we
know, functions very different from the derivatives which have
generated them.
Compared to the implicit differentials, the distinction of orders
is even more important; for, besides the preceding motive,
the influence of which is evidently here analogous, and even to a higher degree,
it is easy to feel that the higher order of the
differential equations necessarily gives rise to questions of a
new nature . Indeed, even if we can integrate indiscriminately any equation
of the first order relating to a single function, this would not be sufficient
to obtain the final integral of an equation of
any order, any differential equation not being reducible to
that of an order immediately lower. If, for example, to
determine a function y of the variable x, we have any relation
between x, y, / frac {dy} {dx}, and / frac {d ^ 2y} {dx ^ 2}, we It will not be possible to
immediately deduce, by carrying out a first integration, the
corresponding differential relation between x, y, and / frac {dy} {dx},
from which, by a second integration, we would go back to the primitive equation.
This would not necessarily take place, at least without introducing new
auxiliary functions, only if the proposed second-order equation did not
contain the sought function y, concurrently with its
derivatives. As a general thesis, differential equations should therefore
really be considered as presenting cases which are all the more
_implicit_ the higher their order is, and which can only fit
into each other by special methods, the
search for which constitutes, for example, therefore, a new class of questions,
about which so far little is known, even for
functions of a single variable [16].
[Note 16: The only important case of this kind which has been
completely treated so far, is the general integration
_linear_ equations of any order,
with constant coefficients. Still, it is found
ultimately to depend on the algebraic resolution of equations of a
degree equal to the order of the differentiation.]
Moreover, when or examines, in a very detailed manner, this
distinction of the various orders of 'differential equations, we find
that it could constantly return to a last
general distinction , relating to differential equations, which I still have to
point out. Indeed, differential equations with one or
more independent variables may contain just
one function, or else,
implicit, which corresponds to the differentiation of
simultaneous implicit functions , one can have to determine at the same time several
functions according to differential equations in which they are
mixed, concurrently with their various derivatives. It is clear that
such a state of the question necessarily presents a new
special difficulty , that of establishing the separation of the different functions
sought, by forming for each, according to the
differential equations proposed, an isolated differential equation, which does not
contain plus the other functions and their derivatives. This
preliminary work , which is the analogue of elimination in algebra, is
Obviously indispensable before attempting any direct integration,
since one cannot generally undertake, unless
special tricks very rarely applicable, to immediately determine
several distinct functions at the same time. Now, it is easy to establish the
exact and necessary coincidence of this new distinction with the
previous one, relating to the order of the differential equations. We know,
in fact, that the general method for isolating functions in
simultaneous differential equations consists essentially in forming
differential equations, separately relating to each function,
and the order of which is equal to the sum of all those of the various equations.
proposed. This transformation can take place constantly. On the other
hand, any differential equation of any order relating to a
single function could obviously always be reduced to first order,
by introducing a suitable number of
auxiliary differential equations , simultaneously containing the various previous derivatives
considered as new functions to determine. This procedure has even
been used with success at times, although in general it is not
normal. There are therefore two kinds of conditions necessarily
equivalent, in the general theory of differential equations, as
the simultaneity of a greater or lesser number of functions, and the order
more or less differentiation of a single function. By
increasing the order of the differential equations, we can isolate all
the functions; and, by artificially multiplying the number of
functions, we can reduce all the equations to the first order. There
is, consequently, in both cases, only the same difficulty,
considered from two different points of view. But, however
one conceives it, this new common difficulty is nonetheless
real, and nonetheless constitutes, by its nature, a
clear separation between the integration of the first order equations and that of the
higher order equations. I prefer to indicate the distinction under
the latter form, as simpler, more general and more
rational.
From the various considerations indicated above on
the rational concatenation of the various principal parts of the
integral calculus , it is seen that the integration of the
explicit differential formulas of the first order with only one variable is the necessary basis
of all the other integrations, that we never succeed in effecting
unless we can fit them into this elementary case, the only one
obviously which, by its nature, is capable of being treated
directly. This simple and fundamental integration is often
referred to by the convenient expression of _quadratures_, since any
integral of this kind Sf (x) dx, can, in fact, be considered as
representing the area of a curve whose equation in
rectilinear coordinates would be y = f (x). Such a class of questions corresponds,
in differential calculus, to the elementary case of the differentiation
of explicit functions with a single variable. But the
integral question is, by its nature, much more complicated, and above all
much more extensive than the differential question. This is
necessarily reduced , as we have seen, to the differentiation
of the ten simple functions, elements of all those which analysis
considers. On the contrary, the integration of compound functions does not
necessarily deduced from that of simple functions, each
new combination of which must present, in relation to integral calculus,
special difficulties. Hence the naturally indefinite extent, and
the so varied complication of the question of quadratures, on which,
despite all the efforts of analysts, we still have so little
complete knowledge.
By breaking down this question, as it is natural to do, according to
the various forms that the derivative function can affect, we
first distinguish the case of algebraic functions, and then that of
transcendent functions . The truly analytical integration of the latter order
of expressions is so far little advanced, either for
exponential functions , or for logarithmic functions, or for
circular functions. We have as yet treated only a very small number of
cases of these three different kinds, choosing them from among the
simplest, which usually lead to extremely
painful calculations . What we must above all remark on this subject
from a philosophical point of view is that the various quadrature procedures do not
derive from any general view on integration, and consist of
simple artefacts of calculation very inconsistent with each other, and of which the number
is greatly multiplied, on account of the very limited extent of each of them. I
I must however point out here one of those artifices which, without
really being a method of integration, is nevertheless remarkable for its
generality: it is the process invented by Jean Bernouilli, and known under
the name of_integration by parts_, of after which any integral
can be reduced to another, which sometimes happens to be
easier to obtain. This ingenious relation deserves to be noted in
another respect, as having offered the first idea of this
transformation into one another of integrals still unknown,
which has recently received a greater extension, and of which M.
Fourier especially has made such a new and important use for
analytical questions generated by the theory of heat.
As for the integration of the _algebraic_ functions, it is more
advanced. However, we still know almost nothing about
irrational functions, the integrals of which have only been obtained
in extremely limited cases, and above all by making them rational.
The integration of rational functions is so far the only theory
of integral calculus which has been able to be treated in a truly
complete way : from the logical point of view, it therefore constitutes the
most satisfactory part, but perhaps also the least. important. It is
even essential to notice, to get a fair idea of the extreme
imperfection of the integral calculus, that this so little extended case is
entirely resolved only for what properly concerns integration,
considered in an abstract manner; for, in the execution, the theory is
most often found, independently of the complication of the calculations,
completely stopped by the imperfection of ordinary analysis, since
it ultimately makes the integration dependent on the
algebraic resolution of equations, which singularly limits its use.
In order to grasp, in a general way, the spirit of the various procedures
according to which one proceeds to the quadratures, we must recognize
moreover that, by their nature, they cannot be founded
initially than on the differentiation of the ten simple functions,
the results of which, considered from the opposite point of view, establish as
many immediate theorems of integral calculus, the only ones which can
be known directly, the whole art of integration then consisting,
as I expressed it at the beginning of this lesson, to make fit, as much
as possible, all the other quadratures in this small number of
elementary quadratures, which unfortunately is still most
often unknown to us.
In this reasoned enumeration of the various essential parts of
integral calculus according to their logical relations, I have
deliberately neglected, in order not to interrupt the sequence, to consider
distinctly a very important theory, which implicitly forms a
portion of the general theory of the integration of
differential equations , but which I must here point out separately, as being,
so to speak, outside the integral calculus, and nevertheless offering the
greatest interest, either by its rational perfection, or by
the extent of its applications. I mean what we call the
_singular_ solutions of differential equations,
sometimes called , but wrongly, _particular_ solutions, which have been the
subject of very remarkable works on the part of Euler and Laplace, and of
which Lagrange especially presented such a beautiful and simple theory
general. We know that Clairaut, who was the first to have the opportunity to
notice its existence, saw in it a paradox of integral calculus, since these
solutions have the specific character of satisfying the
differential equations without being nevertheless included in the
corresponding general integrals . Lagrange has since explained this paradox in
the most ingenious and satisfying way, showing
how such solutions always derive from the general integral
by varying the arbitrary constants. He was also the first to
properly appreciate the importance of this theory, and it is with good
reason that he devoted to it, in his lessons on the calculation of
functions_, such a great development. From the rational point of view,
this theory deserves in fact all our attention, by the character of
perfect generality that it includes, since Lagrange has exposed
invariable and very simple procedures to find the
_singular_ solution of any differential equation whatever it is.
susceptible; and, what is no less remarkable, these procedures
do not require any integration, consisting only in
differentiations, and therefore always applicable. The
differentiation has become, by a happy artifice, a way to
compensate in some circumstances to the imperfection of calculating
integral. Indeed, certain problems require above all, by their nature,
the knowledge of these _singular_ solutions. Such are, for
example, in geometry, all the questions where it is
a question of determining a curve according to any property of its tangent or of its
osculating circle. In all cases of this kind, after having expressed
this property by a differential equation, it will be, in
analytical terms, the _singular_ equation which will constitute the
most important object of the research, since it alone will represent the
requested curve. , the general integral, which therefore becomes useless to
know, not having to designate anything other than the system of tangents
or osculating circles of this curve. One can easily see, from
this, the importance of this theory, which seems to me not
yet sufficiently appreciated by most surveyors.
Finally, to finish pointing out the vast body of
analytical research which constitutes integral calculus proper, it
remains for me to mention a very important theory in all the
applications of transcendent analysis, which I had to leave outside
the system. as not being really intended for a real
integration, and proposing on the contrary to replace the knowledge
of truly analytical integrals, which are most often ignored.
We see that this is the determination of the _defined integrals_.
The expression, always possible, of integrals in indefinite series,
can first of all be considered as a happy general means of
often compensating for the extreme imperfection of the integral calculus. But the use of
such series, because of their complication and the difficulty of
discovering the law of their terms, is usually of mediocre
utility in the algebraic relation, although
very essential relations have sometimes been deduced from them. It is especially under the
arithmetical relation that this process acquires great importance, as a means
of calculating what are called _defined_ integrals, that is to say,
the values of the functions sought for certain determined values
of the corresponding variables.
Research of this nature corresponds exactly, in
transcendent analysis , to the numerical resolution of equations in
ordinary analysis . Not being able most often to obtain the true integral,
that which we call by opposition, the _general_ or _indefinite_ integral,
that is to say, the function which, differentiated, produced the
proposed differential formula , the analysts had to endeavor to determine,
at least without knowing such a function, the
particular numerical values that it would take by assigning values to variables
designated. Obviously, this is solving the arithmetic question, without
having previously solved the corresponding algebraic question, which,
most often, is precisely the most important. Such an analysis
is therefore by its nature, as imperfect as we have seen the
numerical resolution of equations be. It presents, like this one, a
vicious confusion from the arithmetical point of view with the
algebraic point of view ; whence result, either from a purely logical point of view, or
relatively to applications, analogous inconveniences. I can therefore
dispense with reproducing here the considerations indicated in the
fifth lesson on the subject of algebra. It is nevertheless understood that, in
Since it is almost always impossible for us to know the
true integrals, it is of the greatest importance to have been able to
obtain at least this incomplete and necessarily
insufficient solution. Now, this is what we have fortunately achieved today
for all cases, the evaluation of definite integrals having been
reduced to entirely general methods, which leave much to be desired,
in a large number of occasions, only a less complication of
calculations, a goal towards which all the
special transformations of analysts are directed today . Seeing now as
perfect this sort of_transcendent arithmetic_, the difficulty, in
applications, is essentially reduced to making
ultimately the proposed research dependent only on a simple determination
of definite integrals, which, of course, could not always be
possible, whatever analytical skill one may employ to effect such
a forced transformation .
By all the considerations indicated in this lesson, we see
that, if the differential calculus constitutes, by its nature, a
limited and perfect system to which nothing more essential to add, the
integral calculus properly speaking, or the simple treatise on integration,
necessarily presents an inexhaustible field to the activity of the
human mind ,
transcendent is evidently susceptible. The general motives by
which I tried to make one feel, in the fifth lesson,
the impossibility of ever discovering the algebraic resolution of
equations of any degree and of any form, have without a doubt,
infinitely more force still in relation to the search for a
single method of integration, invariably applicable in all cases. _It is_,
says Lagrange, _one of those problems from which we cannot hope for
a general solution_. The more we meditate on this subject, the more we will be
convinced, I am not afraid to say, that such research is
totally chimerical, as being far too superior to the weak.
range of our intelligence, although the work of the surveyors must
certainly increase in the continuation the whole of our
acquired knowledge on integration, and also create procedures of a
greater generality. Transcendent analysis is still too close to its
birth, it is especially too little time that it is conceived in a
really rational way, for us to be able to get a
fair idea of what it could become one day. . But quell es that
must be our legitimate expectations, do not forget to consider
above all the limitations of our intellectual constitution,
and which, though not susceptible to precise determination, not
nonetheless have an indisputable reality.
Instead of tending to imprint on the calculation of indirect functions, as
we conceive it today, a chimerical perfection, I am inclined
to think that when geometers have exhausted the
most important applications of our present transcendent analysis, they will
create themselves. rather new resources, by changing the mode of
derivation of the auxiliary quantities introduced to facilitate
the establishment of the equations, and whose formation could follow an
infinity of other laws than the very simple relation which has been chosen,
according to a design which I have already indicated in the fourth lesson.
Means of this nature appear to me to be susceptible, in themselves,
of greater fertility than those which would consist only of
pushing our present calculation of indirect functions further. It is
a thought that I submit to the geometers whose meditations have
turned to the general philosophy of analysis.
Moreover, although I had, in the summary exposition which was the
proper object of this lesson, to make visible the state of extreme imperfection in which
the integral calculus is still found, one would have a false idea of the
general resources of transcendent analysis, if this
consideration was too great an importance. It is here, indeed, as
in ordinary analysis, where we have succeeded in using, to an
immense degree , a very small number of fundamental knowledge on the
solution of equations. However little advanced they may be
so far in the science of integrations, geometers have nonetheless
drawn, from abstract notions so little multiplied, the solution
of a multitude of questions of prime importance in geometry and
mechanics. , in thermology, etc. The philosophical explanation of this
double general fact results from the
necessarily preponderant importance and scope of abstract knowledge, the
least of which naturally corresponds to a host of research
concrete, man having no other resource for the successive extension
of his intellectual means, than in the consideration of ideas more
and more abstract and nevertheless positive.
To complete the knowledge, in all its extent, of the
philosophical character of transcendent analysis, it remains for me to consider a
last conception by which the immortal Lagrange, which we
find on all the great paths of mathematical science, has
made this analysis even more suitable to facilitate the establishment of
equations in the most difficult problems, by considering a
class of equations even more _indirect_ than the equations
differentials proper. It is the _calcul_ or rather the
_method of variations_, the general appreciation of which will be the subject of
the following lesson.
EIGHTH LESSON.
SUMMARY. General considerations on the calculation of variations.
In order to grasp with greater ease the philosophical character of the
method of variations, it is first necessary to briefly consider
the special nature of the problems whose general resolution required
the formation of this hyper-transcendent analysis. This calculation is still
too close to its origin, its applications have hitherto been too little
varied, for one to be able to conceive of a sufficiently general idea.
clear, if I were to limit myself to a purely abstract exposition of his
fundamental theory, although such an exposition must then,
without doubt, be the main and definitive object of this lesson.
The mathematical questions which gave rise to the _calculus of
variations_ consist, in general, in the search for _maxima_ and
_minima_ of certain indeterminate integral formulas, which
express the analytical law of such or such geometrical or
mechanical phenomenon , considered independently of any subject. particular. The
surveyors appointed for a long time all matters of this
kind by the common name of _problèmes isopérimètres_, which
however suitable for the smallest number of them.
In the ordinary theory of _maxima_ and _minima_, we propose to
discover, relative to a given function of one or
more variables, which particular values should be assigned to
these variables so that the corresponding value of the proposed function
is a _maximum_ or a _minimum_, with respect to those which precede and
which immediately follow, that is to say, one seeks, strictly
speaking, at which moment the function ceases to increase to begin to
decrease, or vice versa. The differential calculus is fully sufficient,
as we know, for the general resolution of this class of questions, in
showing that the values of the various variables which suit, either at the
_maximum_ or at the _minimum_, must always nullify the
different first-order derivatives of the given function, taken
separately with respect to each independent variable; and by indicating
in addition a character suitable for distinguishing the _maximum_ from the _minimum_, which
consists, in the case of a function of a single variable, for example,
in that the derivative function of the second order must take a
negative value for the _maximum_, and positive for the _minimum_. Such,
at least, are the fundamental conditions which relate to the greatest
number of cases; the changes they must undergo in order for the
theory is fully applicable to certain questions, are
moreover also subject to abstract rules which are equally
invariable, although more complicated.
The construction of this general theory having
necessarily removed the main interest which questions of this kind
could inspire in geometers, they rose almost immediately to
the consideration of a new order of problems, both much more
important and of importance. a much greater difficulty, those of the _isoperimeters_.
It is no longer then the values of the variables specific to the _maximum_ or
the _minimum_ of a given function, that it is a question of determining. It's here
form of the function itself that we propose to discover, according to the
condition of the _maximum_ or of the _minimum_ of a certain definite integral,
only indicated, which depends on this function.
The oldest question of this nature is that of the solid of least
resistance, treated by Newton, in the second book of _Principes_, where
he determines what must be the meridian curve of a solid of
revolution, so that the resistance experienced by this body in the direction of
its axis, crossing with any speed an immobile fluid, that
is to say the smallest possible. But the course followed by Newton was
not of a sufficiently simple, fairly general and above all sufficiently
analytical, by the nature of its special method of
transcendent analysis , so that such a solution might suffice to lead
geometers towards this new order of problems. The really
decisive impetus in this regard could hardly have come from one of the surveyors
on the continent busy working out and applying the
infinitesimal method proper. This is what Jean
Bernouilli did in 1695 by proposing the famous problem of brachystochrone, which has
since suggested such a long series of similar questions. It consists in
determining the curve that a heavy body must follow to descend from one
point to another in the shortest time. By limiting ourselves to the simple
fall into a vacuum, the only case that we have first considered, we find
quite easily that the curve sought must be an
inverted cycloid , with a horizontal base, having its origin at the highest point.
But the question can be singularly complicated, either by having
regard to the resistance of the medium, or by taking into account the change
in intensity of gravity.
Although this new class of problems was originally provided
by mechanics, it was nevertheless in geometry that
the subjects of major research were later drawn . Thus, we set out to
discover, among all the curves with the same contour drawn between two
given points, which is the one whose area is a _maximum_ or a
_minimum_, from which properly came the name of _problème des
ipimètres_; or else we asked that the _maximum_ and the _minimum_ take
place for the surface generated by the revolution of the curve
sought around an axis, or for the corresponding volume; in other
cases, it was the vertical height of the center of gravity of the
unknown curve , or of the area and volume it could generate, which
had to become a _maximum_ or a _minimum_, etc. Finally, these problems
were successively varied and complicated, so to speak ad infinitum,
by the Bernouilli, by Taylor, and above all by Euler, before Lagrange
would have subjected the solution to an abstract and entirely
general method, the discovery of which put an end to the eagerness of geometers
for such an order of research. It is not a question here of tracing, even
briefly, the history of this higher part of mathematics,
however interesting it may be. I have only enumerated
certain main questions, chosen from among the simplest, in
order to make clear the general destination which
the method of variations essentially had at its origin.
We see that, considered from the analytical point of view, all these
problems consist, by their nature, in determining which form should
have a certain unknown function of one or more variables,
so that such or such integral dependent on this function is found to
have, between assigned limits, a value which is a _maximum_ or
a _minimum_, relative to all those it would take if the
sought function had any other form. Thus, for example,
in the brachystochrone problem, we know that if y = f (z),
x = / varphi (z) are the rectilinear equations of the sought curve,
assuming the x and y axes horizontal, and the vertical z axis,
the time of the fall of a heavy body along this curve, from
the point whose ordinate is z_1 to the one whose ordinate is z_2
is generally expressed by the definite integral [17]. / [/ int ^ {z_1} _ {z_2}
/ sqrt {/ frac {1+ (f '(z)) ^ 2 + (/ varphi' (z)) ^ 2} {2gz}} dz /.]
[ Note 17: I use the simple and luminous notation
proposed by M. Fourier, to designate the
definite integrals , by clearly mentioning their limits.]
We must therefore find what must be the two unknown functions f
and / varphi for this integral to be a minimum. Likewise, to ask
which is, among all the plane isoperimeter curves, the one which
contains the largest area, is to propose to find, among all
the functions f (x) which can give the integral / [/ int
dx / sqrt {1+ (f '(x)) ^ 2} /] a certain constant value, that which
maximizes the integral / int f (x) dx, caught between the same limits. This
is obviously always the case in all other questions of this
kind.
In the solutions that surveyors gave to these problems before
Lagrange, we essentially proposed to reduce them to the
ordinary theory of maxima and minima. But the means employed to effect
this transformation consisted of simple particular artifices,
specific to each case, and the discovery of which did not involve
invariable and certain rules, so that any really
new question constantly reproduced similar difficulties, without
the solutions already obtained could really be of no
essential help , other than by the habits which they had made
the intelligence contract. In short, this branch of mathematics
then presented the necessary imperfection which constantly exists as long as
we have not succeeded in distinctly grasping, in order to treat it in an
abstract and therefore general manner, the part common to all
questions from the same class.
By seeking to reduce all the various problems of isoperimeters to
depend on a common analysis, organized abstractly into a
distinct calculus , Lagrange was led to conceive of a new nature of
differentiations, to which he applied the characteristic / delta,
reserving the characteristic d for simple
ordinary differentials . These differentials of a new species, which he designated
under the name of _variations_, consist in the
infinitely small increases which the integrals receive, not by virtue
of analogous increases on the part of the corresponding variables,
as in the case of transcendent analysis. ordinary, but assuming that the
form of the function placed under the sign of integration changes
infinitely little. This distinction can be understood, for example, with
ease, relative to the curves, where we see the ordinate or any
another variable of the curve, comprising two kinds of differentials
obviously very different, according to whether one passes from one point to
another infinitely close on the same curve, or else to the
corresponding point of the infinitely close curve produced by a certain
determined modification of the first [18]. It is clear, moreover, that,
by their nature, the relative _variations_ of various magnitudes linked to
each other by any laws, are calculated, except for the characteristic
, in exactly the same way as the differentials. Finally, we
also deduce from the general notion of _variations_ the
fundamental principles of the algorithm specific to this method and which consist
simply in the obvious faculty of being able to transpose at will the
characteristics specially assigned to the variations before or after
those which correspond to the ordinary differentials.
[Note 18: Leïbnitz had already considered the comparison
of a curve to another infinitely close; this is what he
called _différentiatio de curvâ in curvam_. But this
comparison had no analogy with
Lagrange's conception , the Leïbnitz curves being enclosed in the
same general equation, from which they are deduced by the
simple change of an arbitrary constant.]
This abstract conception once formed, Lagrange was able to reduce
easily, in the most general way, all problems from
isoperimeters to the simple ordinary theory of _maxima_ and
_minima_. To get a clear idea of this great and happy
transformation, we must first consider an
essential distinction to which the various questions of
isoperimeters give rise .
We must, in fact, divide this research into two general classes,
depending on whether the _maxima_ and _minima_ requested are _absolute_ or
_relatives_, to use the abbreviated expressions of the surveyors. The
first case is that where the indeterminate definite integrals for which we are
looking for the _maximum_ or the _minimum_, are not subject, by nature
of the problem, under no conditions; as happens, for example, in the
problem of the brachystochrone, where it is a question of choosing between all the
imaginable curves. The second case occurs when, on the contrary, the
variable integrals can change only
under certain conditions, usually consisting in that other
definite integrals , also depending on the functions sought,
constantly keep the same given value; as, for example, in all the
geometrical questions concerning the figures _isoperimeters_ proper
, and where, by the nature of the problem, the integral relating to the
length of the curve or to the area of the surface, must remain constant
during the change of that which is the object of the proposed search.
The calculation of variations immediately gives the general solution of the
questions of the first kind. Because, it obviously follows from the
ordinary theory of _maxima_ and _minima_, that the sought relation must make
null the _variation_ of the integral proposed compared to each
independent variable, which gives the common condition to the maximum and
to the minimum; and, as a characteristic proper to distinguish one from the other, that
the variation of the second order of the same integral must be negative
for the maximum and positive for the minimum. Thus, for example, in the
problem of the brachystochrone, one will have, to determine the nature of the
curve sought, the condition equation, / [/ delta / int_ {z_1} ^ {z_2}
/ sqrt {/ frac {1+ (f '(z)) ^ 2 + (/ varphi' (z)) ^ 2 } {2gz}} dz = 0 /] which, breaking down
into two, with respect to the two unknown functions f and / varphi which are
independent of each other, will completely express the
analytical definition of the requested curve. The only difficulty specific to this
new analysis consists in the elimination of the characteristic
/ delta, for which the calculation of the variations provides
invariable and complete rules , based, in general, on the process of
integration by parts, of which Lagrange has thus knew how to take
immense advantage . The constant goal of this first analytical elaboration, in
the exposition of which I must in no way enter here, is to
arrive at the differential equations properly so called, which can
always be done, and thus the question comes within the domain of
ordinary transcendent analysis , which completes the solution, at least by bringing it back
to pure algebra, if we know how to perform the integration. The
general purpose, specific to the method of variations, is to operate this
transformation, for which Lagrange has established simple,
invariable rules , and of always certain success.
I must not neglect, in this rapid general indication, to
point out, as one of the greatest special advantages of the method of
variations compared to the isolated solutions that we had previously to
isoperimeter problems, the important consideration of what
Lagrange calls the _limit equations_, entirely neglected
before him, and without which nevertheless most of the
particular solutions would necessarily remain incomplete. When the limits
of the proposed integrals must be fixed, their variations being
zero, there is no need to take them into account. But this is no longer the case
when these limits, instead of being strictly invariable, are
subject only to certain conditions; as, for example, if
the two points between which the sought curve must be drawn do not
are not fixed, and should only remain on given lines or
areas. Then, it is necessary to have regard to the variations of their
coordinates, and to establish between them the relations corresponding to the
equations of these lines or of these surfaces.
This essential consideration is only the last complement of a
more general and more important consideration relating to the variations
of the various independent variables. If these variables are really
independent of each other, as when we compare all the
imaginable curves capable of being drawn between two points, it
will be the same for their variations, and consequently the terms relating to
each of these variations must be separately zero in the
general equation which expresses the maximum or the minimum. But if, on the contrary, one
supposes the variables subjected to certain unspecified conditions,
it will be necessary to take into account the relation which results from it between their
variations, so that the number of the equations in which
this general equation is decomposed is then always equal to that
only of the variables which remain truly independent. It is thus,
for example, that instead of looking for the shortest path to go
from one point to another, by choosing among all the possible paths,
one can propose to find only which is the shortest between
all those that can be followed on any given surface, a question
whose general solution certainly constitutes one of the most beautiful
applications of the method of variations.
The problems in which such modifying conditions are considered are
very close, by their nature, to the second general class
of applications of the method of variations, characterized above
as consisting in the search for _relative_ maxima and minima. There
is nevertheless, between the two cases, this essential difference, that, in
the latter, the modification is expressed by an integral which depends on
the sought function, while, in the other, it is designated
by a finite equation which is immediately given. We can see from this
that the search for _relative_ maxima and minima is always and
necessarily more complicated than that of _absolute_ maxima and minima.
Fortunately, a very important general theorem, found before
the invention of the calculus of variations, and which is one of the most beautiful
discoveries due to the genius of the great Euler, gives a uniform and
very simple means of bringing together these two classes of questions. 'one in
the other. It consists, in that if one adds to the integral which must
be a maximum or a minimum a constant and indeterminate multiple of
that which must remain constant by the nature of the problem, it will suffice to
find, according to Lagrange's general procedure, indicated above, the
maximum or the _absolute_ minimum of this total expression. One can
easily conceive, in fact, that the part of the complete variation which
would come from the last integral must as well be zero,
because of the constancy of this one, as the part due to the first
integral, which is annihilated. under the maximum or minimum state. These
two distinct conditions obviously agree to produce, in
this respect, exactly similar effects.
Such is, by outline, the general manner in which the method of
variations applies to all the various questions which compose this
that we called the theory of isoperimeters. We will no doubt have noticed
in this summary exhibition, to what degree was used by
this new analysis the second fundamental property of
transcendent analysis , appreciated in the sixth lesson, namely: the generality
of infinitesimal expressions to represent the same phenomenon
geometric or mechanical, in whatever body is considered. It is,
in fact, on this generality that all
the solutions due to the method of variations are founded by their nature . If a single formula
could not express the length or the area of any curve,
if we did not have another fixed formula to designate the time of the
fall of a heavy body, along whatever line it descends, etc.,
how could it have been possible to resolve questions which
inevitably require , by their nature, the simultaneous consideration of all the
cases which may be determined in each phenomenon by various subjects which
manifest it?
Whatever the extreme importance of the theory of isoperimeters, and
although the method of variations originally had no other object
than the rational and general resolution of this order of problems, we
would still have only an incomplete idea. of this beautiful analysis, if we
limited its destination there. Indeed, the abstract design of two
distinct natures of differentiations, is obviously applicable
not only to the cases for which it was created, but also to all
those which present, by whatever cause, two
different ways of varying the same magnitudes. It is thus that
Lagrange himself made, in his _mécanique analytique_, an immense
capital application of his calculus of variations, by employing it to
distinguish the two kinds of changes which
questions of rational mechanics present so naturally for the various points that we
consider, depending on whether we compare the successive positions occupied,
by virtue of the movement, a same point of each body in two instans
consecutive, or that we pass from one point of the body to another at the
same time. One of these comparisons produces the
ordinary differentials ; the other gives rise to variations, which are, here as
everywhere, nothing but differentials taken from a new point of view.
It is in such a general sense that the calculus
of variations must be conceived , to properly appreciate the importance of this
admirable logical instrument, the most powerful that the human mind has
built up to now.
The method of variations being only an immense extension of the
general transcendent analysis , I do not need to particularly observe
that it is likely to be considered from the various
fundamental points of view involved in the calculation of indirect functions, considered
as a whole. Lagrange invented the calculus of variations according to
the infinitesimal conception proper, and even long before he
undertook the general reconstruction of transcendent analysis. When
he had carried out this important reformation, he easily showed how
it could also be applied to the calculus of variations, which he explained
with all the appropriate development, according to his theory of
derivative functions . But, the more difficult the use of the variation method is
for the intelligence because of the degree of
considered, the more important it is to spare in its application the forces
of our mind, by adopting the most direct and
rapid analytical conception , that is to say, that of Leïbnitz. Also Lagrange himself
he constantly preferred in the important use he made of the
calculus of variations for _mechanics of analysis_. There is not,
in fact, the slightest hesitation in this regard among surveyors.
In order to clarify as completely as possible the
philosophical character of the calculation of variations, I believe I should end by
indicating briefly here a consideration which seems to me important,
and by which I can bring it closer to transcendent analysis.
ordinary to a higher degree than Lagrange seems to me to have
done [19].
[Note 19: I propose to develop this
new consideration later , in a special work on the
_calcul des variations_, which aims to present
the whole of this hyper-transcendent analysis under a
new point of view, which I believe suitable to extend its
general scope.]
We noticed, according to Lagrange, in the previous lesson, the
formation of the calculus with partial differences, created by d'Alembert,
as having introduced, into transcendent analysis, a new idea
elementary, the notion of two distinct kinds of growth and
independent of each other that can receive a function of two
variables, by virtue of changing each variable separately. It is
thus that the vertical ordinate of a surface, or any other magnitude
which relates to it, varies in two quite distinct ways and which
can follow the most diverse laws, sometimes by increasing
the sometimes the other of the two horizontal coordinates. Now, such a
consideration seems to me very close, by its nature, to that which
serves as a general basis for the method of variations. The latter, in fact,
has really done nothing but convey to the
independent variables themselves the way of seeing already adopted for the
fonctions de ces variables, ce qui en a singulièrement agrandi l'usage.
Je crois, d'après cela, que, sous le seul rapport des conceptions
fondamentales, on peut envisager le calcul créé par d'Alembert, comme
ayant établi une transition naturelle et nécessaire entre le calcul
infinitésimal ordinaire et le calcul des variations, dont une telle
filiation me paraît devoir éclaircir et simplifier la notion générale.
D'après les diverses considérations indiquées dans cette leçon, la
méthode des variations se présente comme le plus haut degré de
perfection connu jusqu'ici de l'analyse des fonctions indirectes. Dans
son état primitif, cette dernière analyse s'est présentée comme un
powerful general means of facilitating the mathematical study of
natural phenomena , by introducing, for the expression of their laws, the
consideration of auxiliary quantities chosen in such a way that
their relations are necessarily simpler and easier to
obtain than those of the quantities direct. But the formation of these
differential equations was not conceived as being able to include
any general and abstract rules. Now, the analysis of variations,
considered from the most philosophical point of view, can be
considered as essentially intended, by its nature, to bring
, as far as possible, into the domain of calculation, the establishment
even differential equations, for such is, for a large number of
important and difficult questions, the general effect of
_varied_ equations which, even more _indirect_ than simple
differential equations with respect to the proper objects of research, are
also much easier to be formed, and from which we can then, by
invariable and complete analytical procedures, intended to eliminate the
new order of auxiliary infinitesimals, deduce these
ordinary differential equations, which it would often have been impossible
to establish immediately. The method of variations therefore constitutes
the most sublime part of this vast system of mathematical analysis.
which, starting from the simplest elements of algebra, organizes, by a
succession of uninterrupted ideas, more and more
powerful general means for the in-depth study of natural philosophy, and which,
as a whole, presents , without any comparison, the
most imposing and least ambiguous monument within the reach of the human mind.
But, it must also be recognized that the conceptions usually
considered in the method of variations being, by their nature, more
indirect, more general, and above all much more abstract than
all the others, the use of such a method necessarily requires, and
in a sustained manner, the highest known degree of restraint
intellectual, so as never to lose sight of the precise object of the
research by following arguments which offer to the mind points
of support as little determined, and in which signs are
hardly ever of any help. We must, no doubt, attribute in
large part to this necessary difficulty the little real use that
geometers, except Lagrange, have hitherto made of such an
admirable conception .
NINTH LESSON.
SUMMARY. General considerations on finite difference computation.
The various fundamental considerations indicated in the
previous five lessons really constitute all the essential bases of a
complete exposition of mathematical analysis, considered
from a philosophical point of view. Nevertheless, in order not to neglect any
really important general conception relative to this analysis, I believe it necessary,
before passing to the philosophical study of concrete mathematics, to
explain very briefly the true character proper to a kind of
very extensive calculation, and which , although entering into the substance of
ordinary analysis , is nevertheless still regarded as being of an
essentially distinct nature . This is what is called the _finite
difference calculus_, which will be the special subject of this lesson.
This calculation, created by Taylor, in his famous work entitled _méthodes
incrumentorum_, consists essentially, as we know, in the
consideration of the finite increases which the functions receive as a
result of analogous increases on the part of the
corresponding variables . These increments or _differences_, to which we
apply the characteristic / Delta, to distinguish them from
_differentials_ or infinitely small increments, can, in
their turn, be considered as new functions, and become the subject
of a second similar consideration, and so of continuation, whence results
the notion of the differences of the various successive orders, analogous, at
least in appearance, to the consecutive orders of the differentials. Such
computation obviously presents, like the computation of indirect functions,
two general classes of questions: 1º determining the
successive differences of all the various analytical functions with one or
more variables, as a result of a defined mode of increase of the
independent variables, that we suppose, in general, to increase in
arithmetic progression; 2 ° reciprocally, starting from these
differences, or, more generally, any equations established
between them, go back to the primitive functions themselves, or to their
corresponding relations. Hence the decomposition of this total calculation into
two distinct calculations, to which we usually give the names of
_ direct finite difference calculus_, and _ inverse finite
difference calculus_, the latter also sometimes called _
finite difference integral calculus_. Each of these two calculations would
moreover obviously be susceptible of a rational distribution
similar to that exposed in the seventh lesson for the
differential calculus and the integral calculus, which frees me from making a
separate mention of it.
There is no doubt that, by such a conception, Taylor believed to found
a calculation of an entirely new nature, absolutely distinct from
ordinary analysis, and more general than the calculation of Leibnitz, although
consisting in an analogous consideration. It is also in this way
that almost all surveyors have judged Taylor's analysis. But
Lagrange, with his usual profundity, clearly saw that these
properties belonged more to the forms and notations employed
by Taylor than to the very core of his theory. Indeed, what makes the
specific character of Leïbnitz's analysis, and constitutes it in a
truly distinct and superior calculus , is that the derived functions are,
in general, of an entirely different nature from the primitive functions, in
that they may lead to simpler relations and
training easier, whence result the admirable properties
fundamental to the transcendental analysis, explained in the lessons
previous ones. But this is not the case with the _differences_
considered by Taylor. For these differences are, by their nature,
functions essentially similar to those which generated them,
which makes them unsuitable for facilitating the establishment of equations, and does
not allow them to lead to more general relations.
Any finite difference equation is really, basically, an
equation directly relating to the very quantities of which we compare the
successive states. The scaffolding of new signs, which deceives
the true character of these equations, disguises it however only
in a very imperfect way, since one could always put it
easily demonstrated by constantly replacing the _differences_ by the
equivalent combinations of the primitive quantities, of which they are
really nothing other than the abbreviated designations. Also,
Taylor's calculus has never offered and cannot offer, in any
question of geometry or mechanics, this powerful general help which
we have seen resulting necessarily from Leïbnitz's analysis. Lagrange
, moreover, very clearly established that the alleged analogy observed
between the difference calculus and the infinitesimal calculus was
radically vicious, in the sense that the formulas specific to the first
calculus can in no way provide, as particular cases, those which
are suitable for the second, the nature of which is essentially distinct.
According to the set of considerations which I have just indicated, I believe
that the calculus with finite differences is ordinarily classified wrongly
in the transcendent analysis proper, that is to say in the
calculus of indirect functions. I see it, on the contrary, by
fully adopting the important reflections of Lagrange, which are
not yet sufficiently appreciated, as being only a
very wide and very important branch of ordinary analysis, that is to say, of
what I called the computation of direct functions. Such, in fact,
it seems to me, is its true philosophical character, that the equations which
considers are always, despite the notation, simple
_direct_ equations .
By specifying, as much as possible, the preceding explanation, one must
consider the calculation of Taylor as having constantly for true
object the general theory of the _suites_, of which, before this illustrious
geometer, one had yet considered only the simplest cases .
I should have rigorously mentioned this important theory in
treating, in the fifth lesson, of algebra proper, of which
it is such an extensive branch. But, in order to avoid any duplication,
I preferred to report it only by considering the
finite difference calculation , which, reduced to its simplest general expression, is no other
thing, in all its scope, that a rational study of the
questions relating to the _suites_.
Any _suite_, or succession of numbers deduced from one another
according to any constant law whatever, necessarily gives rise to these
two fundamental questions: 1 ° the law of the sequence being supposed to be
known, find the expression of its general term, so to be able to
immediately calculate a term of any rank, without being obliged
to form successively all the preceding ones; 2º in the same
circumstances, determine the _sum_ of any number of terms in
the sequence as a function of their ranks, so that it can be known
without being forced to continually add these terms to each other.
These two fundamental questions being supposed to be solved, we can
also reciprocally propose to find the law of a series according to
the form of its general term, or the expression of the sum. Each of
its various problems involves all the more extent and difficulty,
as one can conceive of a greater number of different _laws_ for
the series, according to the number of preceding terms on which each term
immediately depends, and according to the function which expresses this
dependence. We can even consider series with several
variable indices , as Laplace did in the analytic theory of
probabilities_, by the analysis to which he gave the name of _theory
of generating functions_, although it is really only
a new and superior branch of the finite difference calculus, or of
the general theory of sequences.
The various general overviews which I have just indicated give even
an imperfect idea of the extent and of the truly infinite variety
of the questions to which geometers have raised themselves from this
sole consideration of series, so simple in appearance, and so limited to
its origin. It necessarily presents as many different cases as the
algebraic resolution of the equations considered in all its extent; and
it is, by its very nature, much more complicated, so much so that
it still depends on it to lead to a complete solution. It is
enough to give a presentiment of what must still be its extreme imperfection, in
spite of the successive works of several geometers of the first order.
We have, in fact, so far only the total and
rational solution of the simplest questions of this nature.
It is now easy to conceive of the necessary and perfect identity
which I announced above, according to the indications of Lagrange, between
the finite difference calculus, and the theory of sequences taken as a
whole. Indeed, any differentiation in the manner of Taylor
obviously comes down to finding the _law_ of forming a sequence with one or
more variable indices, according to the expression of its general term;
in the same way, any analogous integration can be regarded as having for
object the summation of a sequence, whose general term would be expressed by
the proposed difference. In this respect the various problems of calculus
with differences, direct or inverse, solved by Taylor and his
successors, are really of very great value, as they deal with
important questions relative to sequences. But it is very doubtful
whether the form and notation introduced by Taylor really provide
any essential facility in the solution of questions of this kind.
It would be perhaps more advantageous in most cases, and
certainly more rational, to replace the _differences_ by the very
terms of which they designate certain combinations. Since
Taylor's calculation is not based on a truly distinct fundamental thought,
and having its own characteristic only its system of signs, there can be no
real advantage, even in the most favorable assumption,
in conceiving it as detached from it. ordinary analysis, of which it
is, to tell the truth, only an immense branch. This consideration of
_differences_, most often useless when it does not complicate,
seems to me to still retain the character of a time when ideas
Since analytics were not familiar enough to geometers, they
naturally had to prefer special shapes specific to simple
numerical comparisons.
Be that as it may, I must not end this general appreciation
of finite difference calculus without pointing out a new notion to
which it gave birth, and which subsequently assumed great
importance. It is the consideration of these _periodic_ or
_discontinuous_ functions , always keeping the same value for an
infinite series of values subject to a certain law in the
corresponding variables , and which must necessarily be added to the
integrals of the finite difference equations to make them
sufficiently general, as we add simple
arbitrary constants to all the quadratures in order to complete the generality.
This idea, originally introduced by Euler, has become, in
recent times, the subject of very extensive work on the part of M.
Fourier, who transported it into the general system of analysis, and
who made it into a a use so new and so essential for the
mathematical theory of heat that this conception, in its present state,
really belongs to it exclusively.
In order to completely point out the philosophical character of
finite difference calculus , I must not neglect to mention here quickly
the main general applications that have been made so
far.
The solution of questions relating to sequences
should be placed in the first rank, as the most extensive and the most important, if, according to
the explanations given above, the general theory of sequences
were not to be considered as constituting, for example. its nature, the
very basis of Taylor's calculus. This large class of problems being thus
discarded, the most essential of the true _applications_ of
Taylor's analysis , is undoubtedly, until now, the general method of
_interpolations_, so frequently and so usefully employed in the
search for the _empirical_ laws of phenomena. natural. The question
consists, as we know, in intercalating, between certain given numbers,
other intermediate numbers subject to the same law that we suppose to
exist between the first ones. One can fully verify, in this
main application of Taylor's calculus, how much, as I
explained above, the consideration of _differences_ is really
foreign and often troublesome, relative to the questions which depend
on this analysis. Indeed, Lagrange replaced the
interpolation formulas deduced from the ordinary algorithm of
finite difference calculus by much simpler general formulas, which
today are almost always preferred, and which have been found
directly, without playing any role in the superfluous notion of
_differences_, which only complicated the question.
A last important class of applications of
finite difference calculus , which deserves to be distinguished from the preceding one, consists in
the eminently useful use which one makes of it, in geometry, to determine
by approximation the length and the area of any curve whatsoever,
and likewise the quadrature and cubature of a body having
any shape . This process, which can moreover be conceived in the abstract
as depending on the same analytical research as the question of
interpolations, often presents a valuable supplement to the methods
entirely rational geometric models, which frequently lead to
integrations which we do not yet know how to carry out, or to calculations
of a very complicated execution.
Such are the various principal considerations which I thought it necessary to
indicate relative to the computation of finite differences. This examination
completes the philosophical study that I had proposed to sketch for
abstract mathematics. We must now proceed to a
similar work on concrete mathematics, in which we will endeavor above all
to conceive how, supposing the general science of
calculus to be perfect , we have been able, by invariable procedures, to reduce to pure
questions of analysis all the problems that
geometry and mechanics can present , and thus imprint on these two
fundamental bases of natural philosophy, a degree of precision and
above all of unity, in a word, a character of high perfection, which one
such step could only communicate.
TENTH LESSON.
SUMMARY. General view of geometry.
From the general explanation presented in the third lesson
relative to the philosophical character of concrete mathematics,
compared with that of abstract mathematics, I need not
establish here, in a special way, that geometry must to be
considered as a true natural science,
simple and therefore much more perfect than any other. This
necessary perfection of geometry, obtained essentially by
the application, which it so eminently involves, of
mathematical analysis , ordinarily deludes the real nature of this
fundamental science, which most minds today conceive
as a purely rational science, completely independent of
observation. It is nevertheless evident to anyone who
carefully examines the character of geometrical reasonings, even in the
present state of abstract geometry, that if the facts considered therein
are much more interrelated than those relating to any other
science, there always exists, however, in relation to each body studied
by geometers, a certain number of primitive phenomena, which,
not being established by any reasoning, can only be founded on
observation, and constitute the necessary basis of all
deductions. The common error in this respect must be regarded as a
remnant of the influence of the metaphysical spirit, which has so long dominated,
even in geometrical studies. Regardless of its logical gravity,
this false way of seeing continually presents, in the
applications of rational geometry, the greatest drawbacks,
in that it prevents a clear conception of the passage from the concrete to
the abstract.
The scientific superiority of geometry is generally due to the fact that
the phenomena it considers are, necessarily, the most
universal and the simplest of all. Not only can all the bodies of
nature obviously give rise to geometrical research,
as well as to mechanical research, but, moreover,
geometrical phenomena would still exist, even if all the parts of
the universe were supposed to be stationary. . Geometry is therefore, by its
nature, more general than mechanics. At the same time, its phenomena
are simpler; because they are obviously independent of
mechanical phenomena , while these are always necessarily complicated
of the first. It is the same, by comparing geometry with
abstract thermology, which we can conceive today, from the
work of M. Fourier, as I indicated in the third
lesson, as a new general branch of concrete mathematics.
In fact, thermological phenomena, considered even independently
of the dynamic effects which accompany them almost constantly, especially
in fluid bodies, necessarily depend on
geometrical phenomena , since the shape of bodies has a singular influence on the
distribution of heat.
It is for these various reasons that we had to classify previously
geometry as the first part of concrete mathematics, that of
which the study, in addition to its own importance, serves as an indispensable basis for
all the others.
Before directly considering the philosophical study of the various orders
of research which constitute current geometry, it is necessary to form
a clear and exact idea of the general destination of this science,
considered as a whole. This is the object of this lesson.
We commonly define geometry in a very vague and
utterly vicious manner, limiting
ourselves to presenting it as _the science of extension_. It would first be appropriate to improve this definition, by
saying, with more precision, that the object of geometry is
_measurement_ of the extent. But such an explanation would be, in itself,
very insufficient, although it is basically correct. Such an
imperfect overview can in no way convey the true
general character of geometrical science.
To achieve this, I believe I must first clarify two
fundamental notions , which, very simple in themselves, have been singularly
obscured by the use of metaphysical considerations.
The first is that of_space_, which has given rise to so much
sophistic reasoning, to so empty and childish discussions
on the part of metaphysicians. Reduced to its positive acceptation, this
conception consists simply in that, instead of considering
extension in the bodies themselves, we envision it in an
indefinite environment , which we regard as containing all the bodies of
the universe. This notion is naturally suggested to us by
observation, when we think of the_print_ that a
body would leave in a fluid where it would have been placed. It is clear, in fact,
that, from a geometrical point of view, such an _print_ can be
substituted for the body itself, without the reasoning being
altered. As for the physical nature of this indefinite _space_, we
must spontaneously imagine it, for greater ease, as
analogous to the actual environment in which we live, so much so that if this
medium was liquid, instead of being gaseous, our geometrical _space_
would doubtless be conceived also as liquid. This circumstance is
moreover evidently only very secondary, the essential object of such a
conception being only to make us consider extension separately
from the bodies which manifest it to us. We can easily understand _a priori_
the importance of this fundamental image, since it allows us
to study geometrical phenomena in themselves, apart from
all the other phenomena which constantly accompany them in
real bodies, without however exerting on them no influence.
The regular establishment of this general abstraction must be looked at
as the first step which has been taken in the rational study of
geometry, which would have been impossible if it had been necessary to continue to
consider with the shape and the size of bodies the whole of all
their other physical properties. The use of such a hypothesis,
which is perhaps the oldest philosophical conception created by
the human mind, has now become so familiar to us that we have
difficulty in exactly measuring its importance, appreciating the
consequences which would result from it. its removal.
Geometric speculations having thus been able to become abstract, they
have acquired not only more simplicity, but even more
generality. As long as the extent is considered in the bodies themselves,
one can only take as subject of research the forms
effectively realized in nature, which would
singularly restrict the field of geometry. On the contrary, by conceiving of
expanse in_space_, the human mind can envision
any imaginable form, which is essential to give
geometry an entirely rational character.
The second preliminary geometrical conception which we must examine
is that of the different kinds of extent, denoted by the words
_volume_ [20], _surface_, _line_, and even _point_, and of which the
ordinary explanation is so unsatisfactory.
[Note 20: M. Lacroix rightly criticized the expression
of _solide_, commonly used by surveyors to
denote a _volume_. It is certain, in fact, that when
we wish to consider separately a certain portion of
the indefinite_space_, conceived as gaseous, we solidify
by thought the outer enclosure, so that a _line_
and a _surface_ are usually, for our mind,
just as _solid_ as a _volume_. We can even notice
that, more often than not, in order for the bodies to penetrate
each other more easily, we are obliged to
represent to ourselves as hollow the interior of the _volumes_, this
which makes the impropriety of the word
_solid_ still more noticeable .]
Although it is obviously impossible to conceive of any extent
absolutely deprived of any of the three
fundamental dimensions , it is no less indisputable that, on a host
of occasions Geometric questions, even of immediate utility,
depend only on two dimensions, considered separately from the
third, or on a single dimension, considered separately from the
other two . On the other hand, apart from this direct pattern, the study of
one-dimensional and then two-dimensional extension clearly presents itself
as an indispensable preliminary to facilitate the study of bodies.
complete or three-dimensional, the immediate theory of which would be too
complicated. Such are the two general motives which oblige
geometers to consider extent in isolation in relation to one or
two dimensions, as well as in relation to all three
together.
It is in order to be able to think, in a permanent way, of the extension in
two senses or in one only, that the human mind forms the
general notions of _surface_, and of _line_. The hyperbolic expressions
usually used by surveyors to define them tend to lead
to a misconception. But, examined in themselves,
they have no other destination than to allow us to reason with
ease over these two kinds of extent, completely
disregarding what should not be taken into consideration. Now,
for this to happen , it suffices to conceive of the dimension which one wants to eliminate
as gradually becoming smaller and smaller, the other two
remaining the same, until it has reached such a degree of
tenuity that 'she can't fix the attention anymore. It is in this way that one
naturally acquires the real idea of a _surface_, and, by a
second analogous operation, the idea of a _line_, by renewing for the
width what one had first done for the 'thickness. Finally, if we repeat
the same work again, we arrive at the idea of a _point_,
extent considered only in relation to its place, disregarding
any size, and therefore intended to specify
positions. Surfaces obviously have the general property
of exactly circumscribing the volumes; and, likewise the lines, in
turn, circumscribe the surfaces, and are limited by the points. But
this consideration, to which too much importance has often been given,
is only secondary.
Surfaces and lines are therefore really always designed with
three dimensions; it would, in fact, be impossible to represent a
surface other than as an extremely thin plate, and a line
other than like an infinitely loose thread. It is even evident that the
degree of fineness attributed by each individual to the dimensions from which he
wishes to disregard is not constantly the same, for it must
depend on the degree of finesse of his
usual geometric observations . This lack of uniformity has, moreover, no
real drawback , since it suffices, for the ideas of surface and line to
fulfill the essential condition of their destination, that each one
imagine the dimensions to be neglected as smaller than all.
those whose daily experiences give him the opportunity
to appreciate the greatness.
We must probably regret that it is still necessary today
to expressly state an explanation as simple as the preceding one,
in a work such as this one. But I thought I had to
quickly point out these considerations because of the ontological cloud with which a
false way of seeing usually envelops these first notions. We
see by this how devoid of any kind of meaning the
fantastic discussions of metaphysicians on the foundations of
geometry. It should also be noted that these primordial ideas are
usually presented by geometers in a not very
philosophical way , since they expose, for example, the notions of the
different kinds of extent in an order absolutely opposite to their
natural sequence, which often causes
the most serious disadvantages for elementary education .
These preliminaries being posed, we can proceed directly to the
general definition of geometry, always conceiving this science
as having as its final goal the _measurement_ of extent.
In this respect, it is so necessary to enter into a
thorough explanation , based on the distinction of the three kinds of extent, that
the notion of _mesure_ is not exactly the same in relation to
surfaces and volumes as in relation to lines, so that, without
this examination, one would form a false idea of the nature of
geometric questions .
If we take the word _mesure_ in its direct and
general mathematical sense , which simply signifies the evaluation of the _reports_ that
any homogeneous quantities have between them, we must consider, in
geometry, that the _measurement_ of surfaces and volumes , as opposed
to that of lines, is never conceived, even in the
simplest and most favorable cases, as taking place immediately. We
regard the comparison of two lines as direct; that of two
surfaces or two volumes is, on the contrary, constantly indirect. In
fact, it is conceivable that two lines can be superimposed; but the
superposition of two surfaces, or, even more so, that of two
volumes, is obviously impossible to establish in the majority
of cases; and, even when it becomes rigorously practicable,
such a comparison is never neither convenient nor susceptible of exactitude.
It is therefore very necessary to explain what the
truly geometric measurement of a surface or of a volume really consists of.
It is necessary to consider, for this, that, whatever the shape of a
body, there always exists a certain number of lines, more or less
easy to assign, the length of which is sufficient to define exactly the
size of its surface or of its volume. Geometry, considering these
lines as the only ones capable of being measured immediately,
proposes to deduce, from their simple determination, the ratio of the
area or volume sought, to the unit of area or to the unit of
volume. Thus the general object of geometry, relative to surfaces
and volumes, is properly to reduce all comparisons of
surfaces or volumes, to simple comparisons of lines.
In addition to the immense ease that such a
transformation obviously presents for the measurement of volumes and surfaces, it
results, by considering it in a more extensive and more
scientific way, the general possibility of reducing to questions of
lines, all questions relating to volumes and surfaces,
considered as to their size. This is often the most
important use of geometric expressions which determine surfaces and
volumes according to the corresponding lines.
It is not that immediate comparisons between surfaces or between
volumes are never used. But such measures are not
regarded as geometric, and we see in them only a supplement
sometimes necessary, although too rarely applicable, to
the insufficiency or the difficulty of truly rational processes.
It is thus that the volume of a body is often determined, and in
certain cases its surface, from its weight. Likewise, in others
On occasions, when one can substitute for the proposed volume an
equivalent liquid volume , one immediately establishes the comparison of two volumes,
taking advantage of the property which liquid masses present, of being
able to easily take all the forms which one wishes to give them. But
all means of this nature are purely mechanical, and
rational geometry necessarily rejects them.
To make the difference between these determinations more evident with the
true geometric measurements, I will cite a single
very remarkable example , the manner in which Galileo evaluated the ratio of the area of
the ordinary cycloid to that of the generating circle. The geometry of its
time being still too inferior to the rational solution of such a
problem, Galileo imagined to seek this relation by
direct experience . Having weighed as exactly as possible two plates of the same
material and of equal thickness, one of which had the shape of a circle and
the other that of the generated cycloid, he found the weight of the latter
constantly three times that of the cycloid. of the first, from which he concludes that the area
of the cyloid is three times that of the generating circle, a result in
conformity with the true solution obtained later by Pascal and Wallis.
Such a success, on which, moreover, Galileo had not taken the change,
obviously stems from the extreme real simplicity of the relationship sought; and we
conceives the necessary insufficiency of such expedients, even
when they would be effectively practicable.
We can clearly see, from what precedes, of what
properly consist the part of the geometry relating to volumes and that
relating to surfaces. But one does not conceive so clearly the
character of the geometry of the lines, since we seemed, to
simplify the exposition, to consider the measurement of the lines as taking place
immediately. It is therefore necessary, in relation to them, a further
explanation.
To this end, it suffices to distinguish between the straight line and the
curved lines; the measure of the first being alone regarded as
direct, and that of others as constantly indirect. Although the
superposition is sometimes strictly practicable for
curved lines , it is nevertheless evident that truly rational geometry
must necessarily reject it, as not including, even when
possible, any exactitude. The
general object of the geometry of lines is therefore to constantly reduce the measurement of curved lines to
that of straight lines; and consequently, from a more
extended point of view , to reduce to simple questions of straight lines all
questions relating to the size of any curves. To
understand the possibility of such a transformation, we must notice
that in any curve whatever there are always certain
straight lines the length of which must be sufficient to determine that of the
curve. Thus, in a circle, it is evident that from the length of the
radius one must be able to conclude that of the circumference; similarly, the
length of an ellipse depends on that of its two axes; the length
of a cycloid, the diameter of the generating circle, etc .; and if, instead
of considering the totality of each curve, we ask more generally for
the length of any arc, it will suffice to add, to the various
rectilinear parameters which determine the whole of the curve, the chord
of the arc proposed, or the coordinates of its ends. Discover the
The relation which exists between the length of a curved line and that of
similar straight lines, such is the general problem which one has
essentially in view in the part of the geometry relating to the study
of the lines.
By combining this consideration with those previously explained on
volumes and surfaces, we can form a very clear idea
of geometrical science, conceived as a whole, by assigning it
as a general purpose to finally reduce the comparisons of
all species. of extent, volumes, surfaces, or lines, to simple
comparisons of straight lines, the only ones regarded as being able to be
carried out immediately, and which, indeed, could not obviously be
brought back to others easier. At the same time that such a conception
clearly manifests the true character of geometry, it
seems to me suitable for showing, at a single glance, its utility
and perfection.
In order to rigorously complete this fundamental explanation, it
remains for me to indicate how there can be, in geometry, a
special section relating to the straight line, which initially seems incompatible
with the principle that the measurement of this class of lines must to
always be regarded as immediate.
It is, in fact, in relation to that of curved lines, and of all
the other objects that geometry considers. But it is obvious that
the estimation of a straight line can only be considered as direct
insofar as one can immediately relate the linear unit to it. Now,
this is what most often presents insurmountable difficulties,
as I had occasion to explain it especially for another reason
in the third lesson. We must then make the measurement of the
proposed line depend on other similar measurements, capable of being
immediately carried out. There is therefore necessarily a first
distinct geometric study , exclusively devoted to the straight line; its
object is to determine the straight lines, one by the other,
according to the relations proper to any figures resulting from their
assembly. This preliminary part of geometry, which seems,
so to speak, imperceptible when one considers the whole of science, is
nevertheless susceptible of a very great development, when one wishes to
treat it in all its extent. It is obviously all the more
important, since, all the geometric measurements having to be reduced, as
much as possible, to that of the straight lines, the impossibility of
determining the latter would be sufficient to make the solution of
each question incomplete .
Such then, according to their natural sequence, are the various
fundamental parts of rational geometry. We see that for
to follow in its general study a really dogmatic order, it is necessary to
consider first the geometry of the lines, starting with the
straight line , and then to pass to the geometry of the surfaces, to deal
finally that of the volumes. It is no doubt surprising that a
methodical classification which derives so simply from the very nature
of science has not been constantly followed.
After having determined with precision the general and definitive object of
geometrical researches, we must now consider science in
relation to the field embraced by each of its three
fundamental sections .
Thus considered, geometry is obviously susceptible, by its
nature, of a strictly indefinite extension; for the measurement of
lines, surfaces or volumes necessarily presents as many
distinct questions as one can conceive of different forms,
subject to exact definitions, and the number is obviously
infinite.
The geometers at first confined themselves to considering the
simplest forms which nature immediately furnished them, or which were
deduced from these primitive elements by the least
complicated combinations . But they have felt, since Descartes, that, in order to constitute
science in the most philosophical way, it was necessarily
necessary to make it relate, in general, to all forms.
imaginable. They thus acquired the reasoned certainty that this
abstract geometry would inevitably include, as special cases,
all the various real forms that the outside world could
present, so as to never be caught off guard. If, on the contrary,
we had always reduced ourselves to the sole consideration of these
natural forms , without having prepared for it by a general study and by the
special examination of certain simpler hypothetical forms, it is clear that
the difficulties would have were most often insurmountable at the time of
effective application. It is therefore a fundamental principle, in
truly rational geometry, that the need to consider, as much
as possible, all the shapes that can be rigorously designed.
The least thorough examination suffices to show that these forms
present a quite infinite variety. With regard to
curved lines , looking at them as generated by the motion of a point
subject to a certain law, it is clear that we will have, in general, as
many different curves as we will assume different laws
for this motion, which can obviously take place according to an infinity of
distinct conditions, although it can
sometimes happen accidentally that new generations produce curves already
obtained. Thus, to limit myself to only plane curves, if a point
moves so as to remain constantly at the same distance from a
fixed point , it will generate a circle; if it is the sum or the difference of its
distances to two fixed points which remains constant, the curve described
will be an ellipse or a hyperbola; if it is their product, we will have a
completely different curve; if the point always deviates equally from a
fixed point and a fixed line, it will describe a parabola; if it turns
on a circle at the same time as this circle rolls on a straight line, we
will have a cycloid; if it advances along a straight line, while this
straight line, fixed by one of its ends, turns in
any way , it will result in what are generally called spirals
which, by themselves, evidently present as many
perfectly distinct curves , as one can suppose of different relations
between these two motions of translation and rotation, etc., & c.
Each of these various curves can then furnish new ones,
by the various general constructions which geometers have
imagined, and which give rise to evolutes,
to epicycloids, to caustics, etc., etc. Finally, there is obviously an even
greater variety among the double curvature curves.
With respect to surfaces, the forms are necessarily much more
diverse still, considering them as generated by the movement of
lines. Indeed, the form can then vary, not only by
considering, as in the curves, the various laws in
infinite number to which the movement of the
generating line can be subjected , but also by supposing that this line itself comes to
change of nature, which has no analogue in the curves, the
points which describe them cannot have any distinct figure. Two
classes of very different conditions can therefore cause the forms
of surfaces to vary , while only one exists for lines. It
is unnecessary to specifically cite a series of examples suitable for
verifying this doubly infinite multiplicity which one notices among the
surfaces. To get an idea, it would suffice to consider
the extreme variety presented by the only group of so-called
_regulated_ surfaces , that is to say generated by a straight line, and which includes the
whole family of cylindrical surfaces, that of conical surfaces,
the more general class of any involute surfaces, etc. With
regard to the volumes, there is no room for any special consideration,
since they are distinguished from each other only by the surfaces which
terminate them.
In order to complete this geometric overview, it must be added that the
surfaces themselves provide a new general means of designing
new curves, since any curve can be considered as
produced by the intersection of two surfaces. It is in this way, in fact,
that the first lines were obtained, which can be regarded as
really invented by geometers, since nature
immediately gave the straight line and the circle. We know that the ellipse, the
parabola and the hyperbola, the only curves completely studied by
the ancients, had only been conceived, originally, as
resulting from the intersection of a cone with a circular base by a plane
variously located. It is evident that by the combined use of these
different general means for the formation of lines and surfaces,
one could produce a rigorously infinite series of forms.
distinct, starting only from a very small number of figures
directly furnished by observation.
Besides, all the various immediate means for the invention of forms
have hardly any importance since
rational geometry took, in the hands of Descartes, its
definitive character . Indeed, as we will see especially in the twelfth
lesson, the invention of forms is reduced today to the invention of
equations, so that nothing is easier than to conceive
new lines and new surfaces, by changing at will the
functions introduced in the equations. This simple abstract process
is, in this respect,
direct geometric patterns, developed by the most powerful imagination,
which would apply only to this order of designs. It explains
, moreover, in the most general and most sensitive way, the
necessarily infinite variety of geometric forms, which
thus corresponds to the diversity of analytical functions. Finally, it shows no
less clearly that the different shapes of surfaces must be
multiplied even more than those of lines, since lines are
represented analytically by equations with two variables, while
surfaces give rise to equations with three variables, which
necessarily present greater diversity.
The considerations previously indicated suffice to show
clearly the rigorously infinite extension which, by its
nature, comprises each of the three general sections of geometry,
relative to lines, surfaces and volumes, as a result of the
infinite variety of bodies to be measured.
To complete for us an exact and sufficiently extensive idea of the
nature of geometrical research, it is now essential to
come back to the general definition given above, in order to
present it under a new point of view, without which the whole of the
science would only imperfectly designed.
By assigning to geometry the _measurement_ of all kinds of
lines, surfaces and volumes, that is to say, as I have explained,
the reduction of all geometric comparisons to simple
comparisons of straight lines, we obviously have the advantage
of indicating a general destination very precise and very easy to
grasp. But, if setting aside all definition, we examine the
effective composition of geometrical science, we will first be inclined to regard the
preceding definition as much too narrow, for there is no
doubt that the major part of the research which constitutes our
geometry present do not seem to have for object the _measurement_
of the extent. It is probably such a consideration that maintains
again, for geometry, the use of these vague definitions, which
understand everything only because they characterize nothing. I believe
nevertheless, in spite of this fundamental objection, to be able to persist in
indicating the _measurement_ of the extension as the general and uniform goal of
geometrical science, and while including therein all that enters into
its real composition. Indeed, if, instead of confining oneself to considering
the various geometrical researches in isolation, we endeavor to grasp
the main questions, in relation to which all the others,
however important they may be, should be regarded only as
secondary, we will end up recognizing that the _measurement_ of the lines,
surfaces and volumes, is the invariable, sometimes _direct_, and
more often _indirect_, goal of all geometrical work. This
general proposition being fundamental, since it alone can give
our definition all its value, it is essential to enter a
few developments on this subject.
By examining with attention the geometrical researches which do not
seem to relate to the _measurement_ of the extent, we find that
they consist essentially in the study of the various
_properties_ of each line or of each surface, that is to say , in
precise terms, in the knowledge of the different modes of generation,
or at least definition, specific to each form considered.
Now, we can easily establish, in the most general way, the
necessary relation of such a study with the question of _mesure_, for
which the most complete possible knowledge of the properties of
each form is an indispensable preliminary. This is supported
by two equally fundamental considerations, though of a
quite distinct nature.
The first, purely scientific, consists in observing that if one did not
know, for each line or for each surface, any other
characteristic property than that according to which geometers have it.
originally conceived, it would most often be impossible to reach a
solution to the questions relating to its _measurement_. Indeed, it is
easy to feel that the different definitions of which each form is
susceptible are not all equally specific to such a
destination, and that they even present, in this respect,
the most complete oppositions. Now, on the other hand, the
primitive definition of each form obviously not having been able to be chosen according to
this condition, it is clear that one should not generally expect
to find it the most suitable; from which results the need to
discover others, that is to say to study, as much as possible, the
_properties_ of the proposed form. Suppose, for example, that the
circle is defined, the curve which, under the same contour, contains the
largest area, which is certainly a very
characteristic property, one would obviously experience
insurmountable difficulties in deducing from such a starting point the solution of the
fundamental questions relating to the rectification or the squaring
of this curve. It is clear, _a priori_, that the property of having all
its points at an equal distance from a fixed point, must necessarily
adapt much better to research of this nature, without being
precisely the most suitable. Likewise, would Archimedes ever have
discover the quadrature of the parabola, if he had not known of this
curve other property than that of being the section of a cone with a
circular base , by a plane parallel to its generatrix? The purely
speculative works of the preceding geometers, to transform this first
definition, were obviously indispensable preliminaries to the
direct solution of such a question. It is the same, all the more
so, with respect to surfaces. To get a
fair idea, it would suffice to compare, for example, with regard to the question of cubature
or squaring, the ordinary definition of the sphere with that,
no less characteristic no doubt, which would consist in looking at a
spherical body like that which, under the same area, contains the
greatest volume.
I do not need to indicate a greater number of examples to make people
understand, in general, the need to know, as much as possible,
all the properties of each line or of each surface, in order to
facilitate the search for rectifications. , quadratures, and
cubatures, which constitutes the final object of geometry. It can even be
said that the main difficulty with questions of this kind consists in
employing, in each case, the property which best adapts to the
nature of the problem proposed. Thus by continuing to indicate, for greater
precision, the measure of the extent, as the general destination of the
geometry, this first consideration, which goes directly to the bottom
of the subject, clearly demonstrates the need to include in it the study,
as deep as possible, of the various generations or definitions
specific to the same form.
A second reason, of at least equal importance, is that
such a study is essential to organize, in a rational way,
the relation of the abstract to the concrete in geometry.
Geometric science having to consider, as I indicated
above, all conceivable forms which include an
exact definition , it necessarily results, as we have noticed,
are always implicitly included in this abstract geometry,
supposed to have reached its perfection. But when it is necessary to pass
effectively to concrete geometry, one constantly encounters a
fundamental difficulty, that of knowing to which of the various
abstract types one must relate, with a sufficient approximation, the
real lines or surfaces to be studied. . However, it is to
establish such a relation that it is particularly essential to
know the greatest possible number of properties of each shape
considered in geometry.
Indeed, if we always limited ourselves to the only primitive definition
of a line or a surface, even supposing that
_measure_ (which, according to the first kind of considerations, would be
most often impossible), this knowledge would almost
necessarily remain sterile in application, since one would not
ordinarily be able to recognize this form in nature, when it is would
present there . For that, it would be necessary that the uniqueness, according to
which the geometers would have conceived it, was precisely that of which the
external circumstances would involve the verification, a
purely fortuitous coincidence , on which obviously one should not count, although
it could have. sometimes place. It is therefore only by multiplying as
much as possible the characteristic properties of each form
abstract, that we can be assured in advance of recognizing it in
the concrete state, and thus of using all our rational work, by
verifying, in each case, the definition which is capable of being
directly ascertained. This definition is almost always unique in
given circumstances, and varies, on the contrary, for the same form,
with different circumstances: double motive of determination.
Celestial geometry furnishes us, in this regard, with the most
memorable example , well suited to highlighting the general need for
such a study. We know, in fact, that the ellipse was recognized by Kepler
as being the curve described by the planets around the sun and the
satellites around their planets. Now, would this fundamental discovery,
which renewed astronomy have ever been possible, if we
had always confined ourselves to conceiving of the ellipse as the oblique section
of a circular cone through a plane? No such definition could
obviously involve such a verification. The most
common property of the ellipse, that the sum of the distances from all its points to
two fixed points is constant, is doubtless much more likely,
by its nature, to cause the curve to be recognized in this case; but it
is not yet directly suitable. The only character which can
then be verified immediately, is the one which one draws from the relation
which exists in the ellipse between the length of the focal distances and
their direction, the only relation which admits an
astronomical interpretation , as expressing the law which links the distance of the planet
from the sun to the time elapsed since the origin of its revolution. It was therefore
necessary that the purely speculative work of Greek geometers on the
properties of conic sections had previously presented their
generation under a multitude of different points of view, so that
Kepler could thus have passed from the abstract to the concrete, by choosing
among all these various characters the one which could most easily
be observed for the planetary orbits.
I can cite yet another example of the same order, relative to
surfaces, considering the important question of the figure of the earth.
If we had never known any other property of the sphere than its
primitive character of having all its points equally distant from an
interior point , how could we ever have discovered that the surface of the
earth was spherical? For this, it was necessary to deduce
beforehand from this definition of the sphere some properties
likely to be verified by observations made only
at the surface, such as, for example, the constant ratio which exists for the
sphere between the length of the path traveled along a meridian
arbitrary while advancing towards a pole, and the angular height of this
pole on the horizon at each point. It was evidently the same, and
with a much longer series of preliminary speculations, only to
find later that the earth was not strictly spherical,
but that its shape is that of an ellipsoid of revolution.
After such examples, it would undoubtedly be useless to bring back
others, which everyone can easily multiply. We will
always verify there that, without a very extensive knowledge of the various
properties of each form, the relation of the abstract to the concrete in
geometry would be purely accidental, and that, consequently, the
science would lack one of its most essential foundations.
These are therefore the two general reasons which fully demonstrate the
need to introduce into geometry a host of research which does
not have the _measurement_ of extent as its direct object, while continuing, however,
to conceive of such a measure as the final destination of any the
science of geometry. Thus, we can retain the
philosophical advantages presented by the clarity and precision of this
definition, and nonetheless understand in it, in a very rational,
though indirect manner, all the known geometrical researches,
considering those which do not appear to be true. relate to the _measure_ of
the extent, as intended either to prepare the solution of the
final questions , or to allow the application of the solutions obtained.
After having recognized, as a general thesis, the intimate and
necessary relations between the study of the properties of lines and surfaces with
the research which constitutes the final object of geometry, it is
moreover evident that, in the continuation of their work, surveyors
should in no way force themselves to never lose sight of such a
sequence. Knowing, once and for all, how important it is to
vary as much as possible the ways of conceiving each form, they
must continue this study without immediately considering what
utility can be such or such special property for
rectifications, quadratures or cubatures. They would
needlessly hinder their research, attaching childish importance to
the continued establishment of this coordination. The human mind must
proceed, in this respect, as it does on every similar occasion,
when, after having conceived, in general, the destination of a certain
study, it endeavors exclusively to push it as far as possible, in
doing completely disregard this relationship, whose
perpetual consideration complicate its work.
The general explanation that I have just explained is all the more
indispensable, that, by the very nature of the subject, this study of the
various properties of each line and of each surface
necessarily constitutes the very major part of all
geometrical research . In fact, the questions immediately relating to
rectifications, quadratures and cubatures, are obviously, by
themselves, in a very limited number for each form considered. On the
contrary, the study of the properties of the same form presents to
the activity of the human mind a naturally indefinite field, in which one
can always hope to make new discoveries. So, for
example, although geometers have been busy for twenty centuries,
with more or less activity, no doubt, but without any
real interruption , in the study of conical sections, they are far from considering this
subject so simple as exhausted; and it is certain, in fact, that by
continuing to indulge in them, one would not fail to find still
unknown properties of these various curves. If work of this kind
has slowed down considerably over the past century or so, it is not
that it is finished; this is only due, as I will explain later
, to the fact that the philosophical revolution brought about in geometry
by Descartes must have significantly reduced the importance of such
research.
geometry is necessarily infinite because of the variety of shapes
to be considered, but also by virtue of the diversity of points of view from
which the same shape can be considered. This last conception
is even the one which gives the broadest and most complete idea of
all geometric research. We see that studies of this
kind consist essentially, for each line or for each
surface, in relating all the geometrical phenomena which it can
present to a single fundamental phenomenon, regarded as a
primitive definition .
After having exposed, in a general yet precise manner, the
final object of geometry, and showing how science, thus defined,
includes a very extensive class of research which did not
at first seem necessarily to relate to it, it remains for me to consider, as a
whole, the method to be followed for the formation of this science. This
last explanation is essential to complete this first glimpse
of the philosophical character of geometry. I will limit myself for the moment
to indicating the most general consideration in this regard, this
important fundamental notion to be developed and clarified in
the following lessons.
The set of geometrical questions can be treated according to two
methods so different that it results, so to speak, two
kinds of geometries, the philosophical character of which does not seem to me to
have yet been properly grasped. The expressions
_synthetic_ geometry and _analytical_ geometry, usually employed to
denote them, give a very false idea. I would
much prefer the purely historical denominations of _geometry of the
ancients_ and _geometry of the moderns_, which have, at least, the advantage of
not ignoring their true character. But I propose to use
henceforth the regular expressions of _geometry
special_ and _geometry general_, which seem to me suitable to characterize with
precision the true nature of the two methods.
It is not, in fact, in the use of calculus, as is
commonly thought , that the fundamental difference consists precisely between
the way in which we conceive geometry since Descartes, and the
way in which the geometers of antiquity treated
geometric issues . It is certain, on the one hand, that the use of calculus
was not entirely unknown to them, since they made, in their
geometry, continuous and very extensive applications of the theory
of proportions, which was for them as a means of deduction, a sort
of real, though very imperfect and above all extremely limited, equivalent
of our present algebra. We can even use the calculation in a way
much more complete than they did to obtain certain
geometric solutions, which will nevertheless have the essential character
of ancient geometry; this is what happens very frequently, in
relation to these problems of geometry in two or three dimensions,
which are commonly designated under the name of "determined". On the other hand,
however important the influence of calculus may be in our
modern geometry , several solutions, obtained without algebra, can
sometimes manifest the peculiar character which distinguishes it from
ancient geometry , although, in general theory, analysis is essential; I
will cite, as an example, Roberval's method for tangents, of which
nature is essentially modern, and which nevertheless leads, in
certain cases, to complete solutions, without any help from calculation.
It is not therefore by the instrument of deduction employed that we must
principally distinguish the two steps which the human mind can
follow in geometry.
The fundamental difference, so far imperfectly understood, seems to me to
really consist in the very nature of the questions considered. In
effect, geometry, considered as a whole, and supposedly reached
its absolute perfection, must, as we have seen, on the one hand,
embrace every conceivable form, and on the other hand, discover
all the properties of each shape. It is susceptible, according to
this double consideration, to be treated according to two
essentially distinctive planes : either by grouping together all the
questions, however diverse they may be, which concern the same
form, and isolating those relating to different bodies. , whatever
analogy that may exist between them; or, on the contrary, by
bringing together under the same point of view all similar research, to
some diverse forms which they relate to elsewhere, and separating
questions relating to the really different properties of the same
body. In short, the whole geometry can be essentially
ordered or in relation to the studied bodies, or in relation to the phenomena
to be considered. The foreground, which is the most natural, was that of the
ancients; the second, infinitely more rational, is that of the moderns
since Descartes.
Such is, in fact, the principal characteristic of ancient geometry,
that one studied, one by one, the various lines and the various surfaces,
only passing to the examination of a new form when one believed to have
exhausted everything. what interesting shapes could offer
until then. In this way of proceeding, when one undertook
the study of a new curve, the
essential, other than by the geometric exercise to which he had
trained the intelligence. Whatever might be the real similarity of the
questions proposed on two different forms, the
complete knowledge acquired for one could in no way dispense with
repeating the whole of the research for the other. So the course of
the mind was never assured; so that one could not be
certain in advance of obtaining any solution whatever, however analogous
the problem proposed to questions which had already been solved. So, for
example, the determination of the tangents to the three conic sections
provided no rational help in leading the tangent to
some other new curve, like the conchoid, the cissoid, etc. In short
, the geometry of the ancients was, according to the expression proposed
above, essentially _special_.
In the modern system, geometry is, on the contrary, eminently
_general_, that is to say, relative to any form. It is easy
to understand first of all that all geometric questions of
any interest can be proposed in relation to all
imaginable shapes . This is what we see directly for the
fundamental problems , which constitute, according to the explanations given in
this lesson, the final object of geometry, that is to say, the
rectifications, quadratures, and cubatures. But this remark
is no less indisputable, even for research relating to the
various _properties_ of lines and surfaces, and of which the most
essential, such as the question of tangents or tangent planes,
the theory of curvatures, etc., are evidently common to all
forms. The very few searches which are really
peculiar to such and such a form are of extremely
secondary importance . This being said, modern geometry essentially consists in
abstracting, in order to treat it separately, in an entirely general manner,
any question relating to the same geometric phenomenon, in a few
body that it can be considered. The application of the
universal theories thus constructed to the special determination of the phenomenon
in question in each particular body is no longer regarded
as a subordinate work, to be carried out according to invariable rules
and of which the success is certain in advance. . This work is, in a word, of the
same order as the numerical evaluation of a
determined analytical formula . In this respect, there can be no other merit than that
of presenting, in each case, the solution necessarily furnished by the
general method, with all the simplicity and elegance which
the line or the surface considered may have. But we do not attach
of real importance only to the conception and the complete solution of a
new question proper to some form. Works of this
kind alone are regarded as making science take real
steps. The attention of the surveyors, thus exempted from the examination of the
peculiarities of the various forms, and directed entirely to
general questions, could thereby rise to the consideration of
new geometric notions, which, applied to the curves studied by
the elders, have revealed important properties that they
had not even suspected. Such is the geometry since the
radical revolution operated by Descartes in the general system of
science.
The simple indication of the fundamental character proper to each of the two
geometries, is undoubtedly sufficient to demonstrate the immense
necessary superiority of modern geometry. One can even say that
before the great conception of Descartes, rational geometry
was not really constituted on definitive bases, either in the
abstract relation, or in the concrete relation. Indeed, for
science considered speculatively, it is clear that by continuing
indefinitely, as the moderns did before Descartes and even a
little after, to follow the march of the ancients, by adding a few
new curves to the small number of those that they had studied, the
Progress, however rapid they might have been, would have been, after
a long series of centuries, very little in relation to
the general system of geometry, in view of the infinite variety of forms which
have always remained to be studied. On the contrary, with each question
solved following the course of the moderns, the number of
geometric problems to be solved is found, once and for all, reduced
by as much, in relation to all the possible bodies. From a second point
of view, from the complete lack of general methods, it resulted that the
ancient geometers, in all their research, were entirely
abandoned to their own strength, without ever being certain
to get some solution sooner or later. If this imperfection of
science was eminently suited to bringing to light their
admirable sagacity, it must have made their progress extremely slow:
one can get an idea of this from the considerable time they spent
in the study of the sections. conical. Modern geometry,
invariably ensuring the progress of our mind, allows, on the contrary,
to use to the highest possible degree the forces of intelligence, which
the ancients frequently had to consume on questions of very little
importance.
A difference no less capital is manifested between the two systems,
when we come to consider geometry in the concrete relation. In
fact, we noticed earlier that the relation of the abstract to the
concrete in geometry can be firmly founded on bases
rational, unless we directly bring research on
every conceivable form. By studying lines and surfaces
one by one, whatever the number, always necessarily very
small, of those that we have considered, the application of theories
similar to the forms actually existing in nature will not never
that an essentially accidental character, since nothing guarantees that
these forms will be able effectively to return in the abstract types
considered by surveyors.
There is certainly, for example, something fortuitous in
the happy relation which has been established between the speculations of the
Greek geometers on the conical sections and the determination of the
true planetary orbits. By continuing
geometrical work on the same level , we generally had no right to hope
for similar coincidences; and it would have been possible, in these
special studies , that the researches of the geometers would have been directed towards
abstract forms indefinitely inapplicable, while they would have
neglected others, susceptible perhaps of an important and
imminent application . It is clear, at least, that nothing guaranteed
positively the necessary applicability of geometric speculations.
It is quite different in modern geometry. By the sole fact that we
proceed to it by general questions, relating to any form whatsoever,
we have in advance the evident certainty that the forms produced in the
external world could never escape each theory, if the
geometric phenomenon which it envisages comes to appear there.
By these various considerations, we see that the geometry system of the
ancients bears essentially the character of the infancy of science,
which only began to become completely rational as a result of the
philosophical revolution operated by Descartes. But it is evident from a
on the other hand, that geometry could only be conceived at first in this
_special_ way. The _general_ geometry would not have been possible, and
the necessity could not even have been felt, if a long series of
special works on the simplest forms had not
previously provided the bases for Descartes' conception, and made
perceptible the impossibility of persisting indefinitely in the
primitive geometric philosophy .
By specifying as much as possible this last consideration, we must
even conclude that, although the geometry which I have called _general_
must be regarded today as the only true
dogmatic geometry , that to which we will limit ourselves essentially,
the other having, mainly, only a historical interest,
nevertheless it is not possible to make entirely disappear the
_special_ geometry in a rational exposition of science. We
can no doubt dispense, as we have done for about a
century, to borrow directly from ancient geometry all the
results it has provided. The most extensive and
difficult research of which it was composed, are not even
usually presented today except according to the modern method.
But, by the very nature of the subject, it is necessarily impossible to
do without the old method, which, whatever one does,
will always serve dogmatically as a preliminary basis for science, as
it has historically done. The reason is easy to understand. In
fact, the _générale_ geometry was based essentially, as we
would soon establish, on the use of the calculation, the transformation of
geometric considerations in analytical considerations, such a
manner of proceeding can not seize about its immediate
origin . We know that the application of mathematical analysis, by
its nature, can never begin any science whatever,
since it can only take place when science has already been
sufficiently cultivated to establish, relative to the phenomena considered,
some _equations_ which can serve as a starting point for
analytical work . Once these fundamental equations have been discovered, analysis will
make it possible to deduce from them a multitude of consequences, which it would have been
impossible even to suspect at first; it will perfect science to an
immense degree, either as regards the generality of conceptions,
or as regards the complete co-ordination established between them. But, to
constitute the very bases of any natural science whatsoever,
obviously, simple mathematical analysis will never suffice, not
even to demonstrate them again when they have already been founded.
Nothing can dispense, in this regard, from the direct study of the subject,
pushed to the point of finding precise relationships. Trying to
bring science, from its origins, into the domain of calculus
would be to want to impose on the theories bearing on
effective phenomena the character of simple logical processes, and
thus deprive them of all that constitutes their necessary correlation with the
real world . In short, such a philosophical operation, if by itself
it was not necessarily contradictory, can
obviously only end up plunging science back into the domain of metaphysics,
which the human mind has had so much trouble in. come off completely.
Thus, the geometry of the ancients will always, by its nature, have a
first necessary and more or less extensive part in the total system
of geometric knowledge. It constitutes a
rigorously indispensable introduction to _générale_ geometry. But that is
what we must reduce it to in a completely
dogmatic exposition . I will therefore consider directly, in the following lesson,
this _special_ or _preliminary_ geometry, restricted exactly to
its necessary limits, so as not to concern myself more than with the
philosophical examination of the _general_ or _definitive_ geometry, the only
truly rational, and which today essentially composes
science.
ELEVENTH LESSON.
SUMMARY. General considerations on the _special_ or
_preliminary_ geometry .
The geometrical method of the ancients having necessarily to have, according to
the reasons indicated at the end of the preceding lesson, a
preliminary part in the dogmatic system of geometry, to furnish to
the _general_ geometry of the indispensable foundations, it is
now advisable to fix d First of all, in what strictly consists this
preliminary function of the _special_ geometry, thus reduced to the smallest
possible development.
Considering it from this point of view, it is easy to recognize that one
could restrict it to the only study of the straight line for what
concerns the geometry of lines, the squaring of
rectilinear plane areas , and finally the cubature of bodies terminated by
plane faces . The elementary propositions relating to these three
fundamental questions constitute, in fact, the necessary starting point of
all geometrical research; they alone can only be
obtained by a direct study of the subject; while, on the contrary, the
complete theory of all other unspecified forms, even that of the
circle and of the surfaces and volumes which relate to it, can today
come entirely within the domain of _general_ or
_analytical_ geometry , these primitive elements providing already _equations_,
which are sufficient to allow the application of calculation to
geometric questions , which would not have been possible without this precondition.
It follows from this consideration that, in ordinary use, we give
_elementary_ geometry more extent than would be
strictly necessary, since, in addition to the straight line, polygons
and polyhedra, we also include the circle. and _round_ bodies,
the study of which could however be as purely _analytical_ as
that, for example, of conical sections. A thoughtless veneration
for antiquity undoubtedly contributes to maintaining this defect in
method. But as this respect did not prevent us from entering
In the domain of modern geometry the theory of conical sections, it is
necessary that, relatively to circular forms, the
contrary habit , still universal, be founded on other motives. The
most obvious reason that can be given for this is the serious
drawback that there would be, for ordinary education, in postponing until
a period quite remote from mathematical education the solution of
several essential questions, susceptible
immediate and continual application to a host of important uses. To proceed,
in fact, in the most rational way, it would be only with the help of the
integral calculus that we could obtain the interesting results,
relative to the measurement of the length or the area of the circle, or to the
squaring of the sphere, etc., established by the ancients from
extremely simple considerations. This inconvenience would be insignificant
, with regard to minds destined to study the whole of
mathematical science, and the advantage of proceeding with
perfect rationality would, comparatively, be of much greater value. But, the
contrary case being still the most frequent, one had to endeavor to
preserve in elementary geometry properly so called
such essential theories . By admitting the influence of such a consideration,
and no longer restricting this preliminary geometry to what is
Strictly essential, one can even conceive of the utility, for
certain particular cases, of introducing into it several important studies
which have been generally excluded from it, such as those of
conical sections , of the cycloid, etc., in order to include, in a teaching
limited, the greatest possible number of usual knowledge, although,
even in the simple relation of time, it was preferable to follow
the most rational course.
On this subject, I must not take into account here the advantages which
this usual extension of the geometrical method of the
ancients can present beyond the necessary destination which is proper to it, by the
a deeper knowledge which is thus acquired of this method, and by
the instructive comparison which results from it with the modern method. These
are qualities which, in the study of any science,
belong to the course that we have called _historical_, and
which we must know how to give up frankly, when we have clearly recognized
the need to follow the truly _dogmatic_ course. Having
conceived all the parts of a science in the most
rational way, we know how important it is, in order to complete this
education, to study the_history_ of science, and therefore, to
compare exactly the various methods which human mind has
successively employed; but these two series of studies must, in
general, as we have seen, be carefully separated. However, in
the case in question here, the geometric method of the moderns is
perhaps still too recent for it not to be appropriate, in order to
better characterize it by comparison, to deal first, according to the
method of the ancients, certain questions which, by their nature, must
rationally fit into modern geometry.
However that may be, setting aside now these various
accessory considerations , we see that this introduction to geometry, which can
only be treated according to the method of the ancients, is strictly
reducible to the study of the straight line, polygonal areas and
polyhedra. It is even likely that it will end up
usually being restricted to these necessary limits, when the great
analytical notions will have become more familiar, and a study of
the whole of mathematics will be universally regarded as the
philosophical basis of general education. .
If this preliminary portion of geometry, which cannot be
founded on the application of calculus, is reduced, by its nature, to a
series of very little extensive fundamental researches, it is certain,
on the other hand, that it cannot be further restricted, although by a
A real abuse of the analytical mind, in
recent times we have sometimes tried to present from a purely
algebraic point of view the establishment of the principal theorems of
elementary geometry . It is thus that it has been claimed to demonstrate by simple
abstract considerations of mathematical analysis the constant relation
which exists between the three angles of a rectilinear triangle, the
fundamental proposition of the theory of similar triangles, the
measure of rectangles, that parallelepipeds, etc., in a word,
precisely the only geometric propositions which can only be
obtained by a direct study of the subject, without the calculation being
likely to have no part in it. I would not point out here such
aberrations, if they had not been determined by the
evident intention of perfecting, to the highest possible degree, the
philosophical character of geometrical science, by bringing it
immediately, from its birth, into the field of applications of
mathematical analysis. But the capital error committed in this respect by
some geometers must be carefully observed, because it
results from the ill-considered exaggeration of this tendency today
very natural and eminently philosophical, which leads to extend more
and more the influence of analysis in mathematical studies. The
contemplation of the prodigious results which the human mind has
attained in following such a direction, must have involuntarily led
to the belief that even the foundations of concrete mathematics could
be established on mere analytical considerations. It is not,
in fact, for geometry only that we must note
similar aberrations; we shall soon have to observe
perfectly analogous ones relatively to mechanics, on the occasion of the
pretended analytical demonstrations of the parallelogram of forces.
This logical confusion even today has much more gravity in
mechanics, where it actually contributes to spreading a further cloud
metaphysics on the general character of science; while, at
least in geometry, these abstract considerations have hitherto been
left out, without being incorporated into the normal exposition of
science.
According to the principles presented in this work, on
mathematical philosophy , it is not necessary to insist much to make
felt the vice of such a way of proceeding. We have already recognized,
in fact, that calculation being and not being able to be but a means of
deduction, it is to form a radically false idea of it to want
to employ it in establishing the elementary foundations of a
any science ; because, on what would rest, in such an operation,
analytical arguments? A work of this nature, far from
truly perfecting the philosophical character of a science, would
constitute a return to the metaphysical state, by presenting
real knowledge as simple logical abstractions.
When one examines in themselves these alleged
analytical demonstrations of the fundamental propositions of elementary geometry,
one easily verifies their necessary insignificance. They are all
based on a vicious way of conceiving the principle of
homogeneity, the true
general notion of which I explained in the fifth lesson . These demonstrations assume that this principle does not allow
there is no point in admitting the coexistence in the same equation of numbers
obtained by different concrete comparisons, which is
obviously false and visibly contrary to the constant progress of
geometers. Also, it is easy to recognize that by employing the law of
homogeneity in this arbitrary and illegitimate meaning, one could
succeed in _demonstrating_ with just as much apparent rigor
propositions whose absurdity is obvious at first glance. . By
carefully examining, for example, the process by which an
attempt has been made to analytically prove that the sum of the three angles of any right
triangle is constantly equal to two angles
rights, we see that it is founded on this preliminary notion, that if
two triangles have two of their angles respectively equal, the
third angle will also be, on both sides, necessarily equal. This
first point being granted, the relation proposed is deduced
immediately, in a very exact and very simple manner. Now, the
analytical consideration, according to which we wanted to establish this
preliminary proposition, is of such a nature that, if it could be
correct, we would rigorously deduce, by reproducing it in the
opposite direction , this palpable absurdity, that two sides of a triangle
are sufficient, without any angle, for the entire determination of the third
side. Similar remarks can be made on all
demonstrations of this kind, the fallacy of which will thus be verified in a
perfectly sensible manner.
The more we must consider here geometry as being today
essentially analytical, the more it was necessary to protect
minds against this abusive exaggeration of mathematical analysis,
according to which one would claim to dispense with any
geometric observation properly so called, by establishing on pure
algebraic abstractions the very foundations of this natural science. I had to
attach all the more importance to characterizing aberrations thus
linked to the normal development of the human mind, as
so to speak consecrated in recent times by the formal assent
of a very distinguished geometer, whose authority exercises
a very great influence over the elementary teaching of geometry.
I believe I should note on this occasion that, in more than one other
respect, we have, it seems to me, too much lost sight of the character of
natural science necessarily inherent in geometry. It is easy to
recognize this, considering the vain efforts so long attempted by
geometers to _demonstrate_ rigorously, not with the aid of calculation,
but according to certain constructions, several
fundamental propositions of elementary geometry. Whatever we can do, we
Obviously, we cannot avoid resorting sometimes in geometry to
simple immediate observation, as a means of establishing various results.
If, in this science, the phenomena considered are, by virtue
of their extreme simplicity, much more interrelated than those
relating to any other physical science, there must nevertheless
necessarily be some which are not can be deduced, and which,
on the contrary, serve as a starting point. That it is appropriate, as a general thesis,
for the greatest rational perfection of science, to
reduce them to the smallest number possible, that is undoubtedly
incontestable; but it would be absurd to pretend to make them disappear
completely. I admit, moreover, that I find fewer
real disadvantages in extending a little beyond what would be strictly necessary
the number of these geometric notions thus established by
immediate observation , provided they are of sufficient simplicity. , than to
make it the subject of complicated and indirect demonstrations, even when
these demonstrations can be logically irreproachable.
After having characterized as exactly as possible the true
dogmatic destination of the geometry of the ancients reduced to its least
indispensable development, it is advisable to consider briefly as a
whole each of the principal parts of which it must be composed.
I think I can confine myself here to considering the first and most
extensive of these parts, that which has as its object the study of the
straight line ; the other two sections, namely: the squaring of polygons and
the cubature of polyhedra, not being able, given their too
restricted nature , to give rise to any philosophical consideration of any importance,
distinct from those indicated in the previous lesson relative to
the measurement of areas and volumes in general.
The definitive question which one constantly has in view in the study of the
straight line consists properly in determining one by one
the various elements of any rectilinear figure whatever, which makes it possible to
always indirectly know a straight line in whatever
circumstances it may be placed. This fundamental problem is
susceptible of two general solutions, the nature of which is quite
distinct, one graphic, the other algebraic. The first, although
very imperfect, is that which must be considered first, because it
derives spontaneously from the direct study of the subject; the second, much more
perfect in the most important respects, can only be studied
in the last place, because it is founded on the
prior knowledge of the other.
The graphic solution consists in _reporter_ at will the
proposed figure , either with the same dimensions,
dimensions varied in any proportion. The first mode
can hardly be mentioned except for the record, as being the simplest,
and that which the mind must consider first, for it is, of course,
moreover almost entirely inapplicable by its nature. The second
is, on the contrary, susceptible of the most extensive and
useful application. We still make extensive and
continual use of it today , not only to represent exactly the shapes of
bodies and their mutual positions, but even for the
effective determination of geometric quantities, when we do not need
great precision. . The ancients, given the imperfection of their
geometric knowledge, employed this process in a
much more extensive manner, since it was for a long time the only one that they could
apply, even in the most important precise determinations.
Thus, for example, Aristarchus of Samos estimated the
relative distance of the sun and the moon to the earth, by taking measurements on
a triangle constructed as exactly as possible so as to be
similar to the right triangle formed by the three stars, at the moment
when the moon is in quadrature, and where, consequently, it would be enough,
to define the triangle, to observe the angle to the earth. Archimedes
himself, although he was the first to introduce into geometry the
calculated determinations, has several times employed similar means.
The formation of trigonometry has not even caused it to be
entirely renounced , although it has greatly reduced its use; the Greeks and
the Arabs continued to use it for a multitude of researches, in which
we now regard the use of calculus as indispensable.
This exact reproduction of any figure on a
different scale can present no great theoretical difficulty when
all the parts of the proposed figure are included in the same
plane. But, if we suppose, as it happens most often, that they
are situated in different planes, we then see the birth of a new
order of geometric considerations. The artificial figure, which is
constantly flat, no longer being able, in this case, to be a
perfectly faithful image of the real figure, it is first necessary to fix with
precision the mode of representation, which gives rise to the various
systems of _projection_. That said, it remains to determine according to
which laws the geometric phenomena correspond in the two
figures. This consideration engenders a new series of
geometrical researches , the final object of which is properly to discover how
we can replace constructions in relief by
plane constructions . The elders had to solve several basic questions
of this kind, for the various cases in which we now employ
spherical trigonometry; and mainly for the various problems
relating to the celestial sphere. Such was the destination of their
_analemnes_, and of the other plane figures which for so
long supplied the use of calculus. We see from this that the ancients
really knew the elements of what we now call
_descriptive geometry_, although they did not conceive them in a
distinct and general manner.
I think it appropriate to point out here quickly, on this occasion, the
true philosophical character of this descriptive geometry, although
that, as being a science essentially of application, it
should not be understood in the proper domain of this work, as
I circumscribed it at the beginning.
All questions of three-dimensional geometry
necessarily give rise, when one considers their graphic solution,
to a general difficulty which is peculiar to them, that of substituting for the
various constructions in relief necessary to solve them, and which
are almost always of necessity . an impossible execution, simple
equivalent plane constructions, capable of finally determining
the same results. Without this essential conversion, each
A solution of this kind would obviously be incomplete and really
inapplicable in practice, although, for the theory,
constructions in space are usually preferable as more
direct. It is in order to provide the general means of
constantly carrying out such a transformation that _descriptive geometry_
was created, and constituted into a distinct and homogeneous body of doctrine
by a view of the genius of our illustrious Monge. He previously devised
a uniform mode of representing bodies by figures drawn on a
single plane, with the aid of _projections_ on two different planes,
usually perpendicular to each other, and one of which is supposed
turn around their common intersection to merge with
the extension of the other; it was enough, in this system, or in any
other equivalent, to look at the points and the lines, as determined
by their projections, and the surfaces by the projections of their
generatrices. That being said, Monge, analyzing with profound sagacity the
various partial works of this kind carried out before him according to a host
of incoherent procedures, and even considering, in a general and
direct manner, in what must constantly consist the questions
whatever of this nature, recognized that they were always
reducible to a very small number of invariable abstract problems,
capable of being solved separately once and for all by
uniform operations, and which relate primarily to the
contacts and the others to the intersections of surfaces. Having formed
simple and entirely general methods for the graphic solution of
these two orders of problems, all the geometric questions to
which the various arts of
construction, the cutting of stones, the framework, the perspective, the
gnonomonics, the fortification, can give rise , etc., could henceforth be treated
as simple particular cases of a single theory,
the invariable application of which will always necessarily lead to a solution
accurate, which can be facilitated in practice by taking advantage of the
specific circumstances of each case.
This important creation singularly deserves to fix the attention of
all the philosophers who consider the whole of the operations of
the human species, as being a first step, and so far the only
really complete, towards this general renovation of human works,
which must impart to all our arts a character of precision and
rationality, so necessary for their future progress. Such a revolution
was, in fact, inevitably to begin with that class of
industrial work which relates essentially to the simplest science,
the most perfect, and the oldest. It cannot fail to
extend successively in the following, although with less facility,
to all the other practical operations. We shall even soon have
occasion to observe that Monge, who conceived
the true philosophy of the arts more deeply than anyone else, had tried to outline for
the mechanical industry a doctrine corresponding to that which he had so
happily formed for the geometric industry. , but without obtaining for
this case, the difficulty of which is much greater, no other success than
that of indicating clearly enough the direction which
research of this nature should take .
However essential the conception of
descriptive geometry may really be , it is very important not to misunderstand the true
destination which is so expressly peculiar to it, as did,
especially in the early days of this discovery, those who there saw
a way to enlarge the general and abstract field of
rational geometry . The event has in no way responded to these
ill-conceived hopes since . And, in fact, is it not evident that
descriptive geometry has special value only as a science of application,
as constituting the true proper theory of the geometric arts?
Considered in the abstract relation, it cannot introduce any
really distinct order of geometric speculations. We must not
lose sight of the fact that, for a geometrical question to fall within the
proper domain of descriptive geometry, it must necessarily
have always been resolved beforehand by speculative geometry, of
which then, as we have seen, the solutions constantly
need to be prepared for practice in such a way as to supplement the
relief constructions by plane constructions, a substitution which
really constitutes the only characteristic function of
descriptive geometry .
It should nevertheless be noted here that, from the point of view of
intellectual education, the study of descriptive geometry presents
an important philosophical property, quite independent of its
high industrial utility. This is the advantage which it offers so
eminently, by accustoming to consider in space
geometrical systems which are sometimes very compound, and to follow exactly their
continual correspondence with the figures actually traced,
to thus exercise in the highest degree in the surest and most
precise manner, that important faculty of the human mind which is called
the_imagination_ properly so called, and which consists, in its
elementary and positive acceptation , in representing clearly and easily a
vast variable set of fictitious objects, as if they were right in front of our
eyes.
Finally, to complete the indication of the general nature of
descriptive geometry by determining its logical character, we must observe
that if, by the kind of its solutions, it belongs to the geometry
of the ancients, on the other hand it approaches the geometry of the
moderns by the species of the questions which compose it. These questions
are, in fact, eminently remarkable for this generality which we
saw, in the last lesson, constitute the true
fundamental character of modern geometry; the methods are always
conceived there as applicable to any forms, the particularities
specific to each form can only have a purely influence.
secondary. The solutions are therefore graphic like most of
those of the ancients, and general like those of the moderns.
After this important digression,
the necessity of which the reader will no doubt have recognized, let us continue the philosophical examination of
_special_ geometry , considered always as reduced to its least
possible development, to serve as an indispensable introduction to
_general_ geometry. Having sufficiently considered the graphic solution
of the fundamental problem relating to the straight line, that is to say, of the
determination by one another of the various elements of
any rectilinear figure , we must now examine it in a way.
general the algebraic solution.
This second solution,
the obvious superiority of which it is useless here to expressly appreciate , necessarily belongs, by the very nature
of the question, to the system of ancient geometry, although the
logical process employed usually makes it inappropriate to separate from it. We thus
have reason to verify, in a very important respect, what has
been established in general in the preceding lesson, that it is not by
the use of calculus that we must essentially distinguish
modern geometry from that of old. The ancients are, in fact, the real
inventors of current trigonometry, both spherical and rectilinear,
which only was much less perfect in their hands, seen
the extreme inferiority of their algebraic knowledge. It is therefore
really in this lesson, and not, as one might believe
at first, in those which we will then devote to the
philosophical examination of _general_ geometry, that the
character of this important theory should be appreciated. preliminary, usually
wrongly included in what is called _geometry analytic_, and which
is effectively only a complement of _ elementary geometry_
proper.
Since all rectilinear figures can be broken down into triangles,
it is obviously sufficient to know how to determine the
various elements of a triangle by one another,
simple _trigonometry_.
So that such a question can be solved algebraically, the
difficulty consists essentially in forming between the angles and the
sides of a triangle three distinct equations, which, once obtained,
will obviously reduce all trigonometric problems to pure
calculus research. By considering
the establishment of these equations in the most general way , we immediately see the birth of a
fundamental distinction relative to the mode of introduction of the angles
into the calculation, according to whether they are made to enter it directly by themselves
or by the circular arcs which are proportional to them, or that, at
On the contrary, certain straight lines will be substituted for them, such as, for example,
the strings of these arcs which are inherent to them, and which, for this
reason, are ordinarily called their trigonometric lines. Of these
two systems of trigonometry, the second must have been, originally, the
only one adopted, as being the only practicable, since the state of the
geometry then made it possible to find quite easily
exact relations between the sides of the triangles and the trigonometric lines of the
angles, while it would have been absolutely impossible at that time
to establish equations between the sides and the angles themselves. The
solution can today be obtained indifferently in one and
in the other system, this pattern of preference no longer exists. But the
geometers none the less had to persist in following by choice the system
originally admitted by necessity; because, the same reason which
thus made it possible to obtain the trigonometric equations with much more
facility, must also, as it is even easier to conceive it
_a priori_, to make these equations much simpler, since they
then exist only between straight lines, instead of being
drawn between straight lines and arcs. Such a
consideration is all the more important since these are
eminently elementary formulas , intended to be continually employed in
all parts of mathematical science as well as all of
its various applications.
One can object, it is true, that, when an angle is given, it is
always in effect by itself and not by its trigonometric line; and
that, when it is unknown, it is its angular value that it is
properly to determine, and not that of any of its
trigonometric lines . It seems, from this, that such lines are
between the sides and the angles only a useless intermediary, which must
be finally eliminated, and the introduction of which does not seem
likely to simplify the research which one proposes. . It is important, in
fact, to
Usually makes immense real utility in this way of proceeding.
It consists in the fact that the introduction of these auxiliary quantities
divides the total question of trigonometry into two other
essentially distinct ones, one of which has for object to pass angles
to their trigonometric lines or vice versa, and of which the other
proposes determine the sides of triangles by the
trigonometric lines of their angles or vice versa. Now, the first of
these two fundamental questions is obviously capable, by its
nature, of being entirely treated and reduced to numerical tables
once and for all, by considering all possible angles, since it
depends only on these angles, and in no way on the particular triangles into which
they may enter in each case; while the solution of the second
question must necessarily be repeated, at least in the
arithmetical respect , with each new triangle that must be solved. This is
why the first portion of the total work, which would be precisely the
most painful, is usually no longer counted, being always done
in advance; while if such a decomposition had not been
instituted, one would obviously have found oneself in the obligation to
recommence in each particular case the entire calculation. This is
the essential property of the trigonometric system adopted, which, in
Indeed, would not present any effective advantage if, for
each angle to be considered, its
trigonometric line had to be calculated continuously or vice versa: the intermediary would then be more
inconvenient than convenient.
In order to understand clearly the true nature of this conception, it
will be useful to compare it with an even more important conception,
intended to produce a similar effect, either under the algebraic relation,
or especially under the arithmetical relation, the admirable theory of
logarithms. . By examining in a philosophical way the influence of
this theory, we see, in fact, that its general result is to have
decomposed all the arithmetic operations imaginable in two
distinct parts, of which the first, which is the most complicated, is
capable of being executed in advance once and for all, as
depending only on the numbers to be considered and not at all on the various
combinations whatever into which they may enter, and which
consists in representing all the numbers as
assignable powers of a constant number; the second part of the calculation, which must
necessarily be restarted for each new formula to be evaluated,
being therefore reduced to performing on these exponents
infinitely simpler correlative operations . I limit myself to indicating this
connection, which everyone can easily develop.
We must also observe as a property, secondary
today, but capital at the origin, of the
adopted trigonometric system , the very remarkable circumstance that the determination of the angles
by their trigonometric lines or vice versa, is susceptible
of an arithmetical solution. , the only one which is directly essential
for the proper destination of trigonometry, without having previously
solved the corresponding algebraic question. It is undoubtedly to
such a peculiarity that the ancients must have been able to know
trigonometry. The research thus conceived was all the easier since,
the ancients having naturally taken the rope for line
trigonometric, the tables were found to have been built
in advance in part for a completely different reason, by virtue of Archimedes' work on
the rectification of the circle, from which resulted the effective determination
of a certain series of strings, so that, when
Hipparchus later invented trigonometry, he could limit himself to completing
this operation by suitable intercalations, which
clearly marks the lineage of ideas in this regard.
In order to fully sketch this philosophical outline of
trigonometry, it should now be observed that the extension of the same
motif which leads to the replacement of angles or arcs of circles by
straight leagues in order to simplify the equations, must also lead
to the concurrent use of several trigonometric lines,
instead of limiting oneself to one, as the ancients did, to
perfect this system by choosing the one which will be algebraically the
most suitable in such a way or such occasion. In this respect, it is
clear that the number of these lines is in itself by no means limited;
provided that they are determined according to the arc, and that reciprocally
they determine it, according to whatever law they derive
from it, they are apt to be substituted for it in the equations.
By limiting themselves to the simplest constructions, the Arabs and
Moderns then successively increased to four or five the number
of _direct_ trigonometric lines, which could be extended much
further. But, instead of having recourse to geometric formations which
would end by becoming very complicated, we conceive with extreme
facility as many new trigonometric lines as
the analytic transformations may require, by means of a
remarkable artifice , which is not not usually understood in a fairly
general way . It consists, without immediately multiplying the
trigonometric lines specific to each arc considered, to introduce
new ones by looking at this arc as determined indirectly by all
the lines relating to an arc which is a very simple function of the
first. It is thus, for example, that often, in order to calculate an angle
with greater ease, one will determine, instead of its sine, the sine of
its half or of its double, etc. Such a creation of
_indirect_ trigonometric lines is obviously much more fruitful than all
the immediate geometrical processes to obtain new ones. We
can say, from this, that the number of trigonometric lines
actually employed today by geometers is really
indefinite, since, at any moment so to speak, the
analytical transformations can lead to increasing it by the process I come
to indicate. Only, so far, special names have only been given to
those of these _indirect_ lines which relate to the complement of
the primitive arc, the others not recurring enough frequently to
require similar names, which has commonly done
misunderstand the true extent of the trigonometric system.
This multiplicity of trigonometric lines obviously gives rise to
a third fundamental question in trigonometry, the study of the
relations which exist between these various lines; since, without
such knowledge, one could not use, for
analytical needs , this variety of auxiliary quantities, which however
no other destination. It is clear, moreover, from the
consideration indicated above, that this essential part of
trigonometry, although simply preparatory, is, by its nature,
susceptible of indefinite extension when considered in its
entire generality, while the other two are necessarily
circumscribed within a strictly defined framework.
I need not add expressly that these three
main parts of trigonometry must be studied in
precisely the reverse order of that in which we have seen them
necessarily derive from the general nature of the subject; because the third is
visibly independent of the other two, and the second from that which
arose first, the resolution of triangles proper
, which must, for this reason, be treated last, which
made the consideration of
natural parentage .
It was useless to consider
spherical trigonometry here separately , which cannot give rise to any
special philosophical consideration , since, however essential it may be by the importance and
the multiplicity of its uses, it can no longer be treated today. hui,
as a whole, than as a simple application of
rectilinear trigonometry , which immediately provides its fundamental equations, in
substituting the corresponding trihedron angle for the spherical triangle.
I thought I should indicate this summary exposition of
trigonometric philosophy , which could moreover give rise to many other
interesting considerations, in order to make visible, by an
important example , this rigorous sequence and this successive ramification
presented by the questions. the simplest in appearance of
elementary geometry.
Before thus sufficiently considered for the purpose of this work the
specific character of the _special_ geometry, reduced to its only
dogmatic destination, of providing the _general_ geometry with an
indispensable preliminary basis , we must henceforth carry all our
attention to true geometrical science, viewed as a
whole in the most rational way. To this end, we must first
carefully examine Descartes' great mother idea, on
which it is entirely based, which will be the subject of the
next lesson .
TWELFTH LESSON.
SUMMARY. Fundamental conception of _general_ or
_analytical_ geometry .
Since _general_ geometry is entirely based on the transformation
of geometric considerations into
equivalent analytical considerations , we must first examine directly and in
depth the beautiful conception according to which Descartes established
uniformly the constant possibility of such a correlation. Besides its
own extreme importance, as a means of eminently perfecting
geometrical science, or rather of constituting it as a whole on
rational bases, the philosophical study of this admirable
conception must have in our eyes an interest all the more elevated, since 'it
characterizes with perfect evidence the general method to be used
to organize the relations from the abstract to the concrete in mathematics,
by the analytical representation of natural phenomena. There is
no thought in mathematical philosophy which deserves more
attention.
In order to be able to express by simple analytical relations all
the various geometrical phenomena that one can imagine, it is
obviously necessary first to establish a general mode to represent
analytically the very subjects in which these phenomena reside,
that is - say the lines or surfaces to be considered. The _subject_ being
thus usually considered from a purely analytical point of view, it
is understood that from then on it has been possible to conceive in the same
way any _accidens_ of which it is susceptible.
To organize the representation of geometric shapes by
analytical equations, one must first overcome a difficulty
fundamental, that of reducing the
general elements of the various geometrical notions to simply numerical ideas; in a word, to
substitute, in geometry, pure considerations of _quantity_ for all
considerations of _quality_.
To this end, let us first observe that all geometrical ideas
necessarily relate to these three universal categories: the
size, the shape and the position of the extents to be considered. As for the
first, there is obviously no difficulty; it
immediately returns to the ideas of numbers. For the second, it should be
noted that it is always reducible by its nature to the third.
For the shape of a body obviously results from the mutual position of
different points of which it is composed, so that the idea of position
necessarily includes that of form, and that any circumstance of
form can be translated by a circumstance of position. This is
indeed how the human mind proceeded to arrive at the
analytical representation of geometric forms, the conception being
directly relative only to positions. All the elementary difficulty
is therefore properly reduced to reducing any ideas of situation to
ideas of grandeur. Such is the immediate destination of the
preliminary conception on which Descartes established the
general system of analytical geometry.
His philosophical work has simply consisted, in this respect, in
the entire generalization of an elementary process which can be regarded
as natural to the human mind, since it is formed, so to speak,
spontaneously in all intelligences, even more vulgar. In
fact, when it comes to indicate the position of an object without the
show immediately, the way we always adopt, and the only
course that can be used, is to bring this item to
others who are known , by assigning the magnitude of
any geometric elements , by which it is conceived linked to these [21].
These elements constitute what Descartes, and according to him all the
geometers, called the _coordinates_ of each point considered, which
are necessarily two in number if we know in advance in which
plane the point is located, and three in number, if it can be found
indifferently in a region any of space. As many
different constructions one can imagine to determine the position
of a point, either on a plane or in space, as many
different coordinate systems are conceived , which are therefore liable
to be multiplied by the infinite. But whatever the system
adopted, we will always have reduced the ideas of situation to simple
ideas of magnitude, so that we can imagine the displacement of a
point as produced by pure numerical variations in the values
of its coordinates. To first consider only the least
complicated case, that of plane geometry, it is thus that we
most often determine the position of a point on a plane, by its more
or less large distances on two lines. fixed supposedly known, which we call
_axes_, and which we usually suppose to be perpendicular to each other. This
system is the most adopted, because of its simplicity; but geometers
sometimes still employ an infinite number of others. Thus, the
position of a point on a plane can be determined by its distances to
two fixed points; or by its distance to a single fixed point, and the
direction of this distance, estimated by the greater or lesser angle
it makes with a fixed line, which constitutes the
so-called _polar_ coordinate system, the most commonly used after that of which we
spoke first; or by the angles formed by the lines going from the
variable point to two fixed points with the line joining the latter; or
by the distances from this point to a fixed line and a fixed point, etc.
In short, there is no geometrical figure whatsoever from which one can
not deduce a certain system of coordinates, more or less
likely to be used.
[Note 21: This is so, for example,
distances greater or lesser to the equator and to a prime
meridian.]
A general observation which it is important to make in this regard is that
any system of coordinates amounts to determining a point, in
plane geometry, by l 'intersection of two lines, each of which is
subject to certain fixed conditions of determination; only one of
these conditions remaining variable, and sometimes one, sometimes another,
depending on the system considered. We cannot, in fact, conceive of any other
means of constructing a point than to mark it by the meeting of
any two lines. Thus, in the most frequent system, that of
_ rectilinear coordinates_ proper, the point is determined by
the intersection of two straight lines, each of which remains constantly parallel
to a fixed axis, moving away from it more or less; in the
_polar_ system , it is the meeting of a circle of variable radius and the
center of which is fixed, with a movable line subject to rotate around this
center, which marks the position of the point; by choosing other
systems, the point could be designated by the intersection of two
circles, or any two other lines, etc. In short, assigning
the value of one of the coordinates of a point in whatever system it
may be is always necessarily to determine a certain line
on which this point must be located. The geometers of antiquity
had already made this essential remark, which served as the basis of their
method of the _geometric places_, of which they made such happy use
to direct their research in the resolution of
_determined_ geometry problems, by separately appreciating the influence of each
of the two conditions by which was defined each point constituting
the object, direct or indirect, of the proposed question: it is precisely
this method whose general systematization was for Descartes the
immediate motive of the works which led him to found
analytical geometry .
Having clearly established this preliminary conception, by virtue of
which the ideas of position, and, consequently implicitly, all the
elementary geometrical notions, are reducible to simple
numerical considerations, it is easy to conceive directly, in
its entire generality, the great mother idea of Descartes, relating to the
representation analysis of geometric shapes, which is
the proper subject of this lesson. I will continue to consider initially,
for the sake of ease, only two-dimensional geometry, the only one that
Descartes treated, then having to examine separately from the same
point of view what is specific to the theory of surfaces or curves.
double curvature.
From the way of analytically expressing the position of a point on
a plane, one can easily establish that, by whatever property
any line may be defined, this definition is always
susceptible to being replaced by a corresponding equation between the
two variable coordinates of the point which describes this line, equation which
will therefore be the analytical representation of the proposed line, of which
any phenomenon will have to result in a certain
algebraic modification of its equation. If we suppose, in fact, that a point
moves on a plane without its course being determined in any way,
we will obviously have to look at its two coordinates, in some system
be it, as two variables entirely independent of
each other. But, if, on the contrary, this point is subject to describing any
certain line, it will necessarily be necessary to conceive that its
coordinates retain between them, in all the positions that it can
take, a certain permanent and precise relation, susceptible,
consequently, of be expressed by a suitable equation, which will become
the very clear and very rigorous analytical definition of the line
considered, since it will express an algebraic property exclusively
relative to the coordinates of all the points of this line. It is
clear, in fact, that when a point is not subject to any condition, its
situation is determined only insofar as one gives at the same time its two
coordinates, distinctly one from the other; while when the point
must lie on a defined line, a single coordinate is sufficient to
fully fix its position. The second coordinate is then then a
determined _function_ of the first, or, in other words, there must
exist between them a certain _equation_, of a nature corresponding
to that of the line on which the point is subject to remain. In short
, each of the coordinates of a point obliging it to be located on a
certain line, we reciprocally understand that the condition, on the part
of a point, of having to belong to a defined line of
arbitrary, is equivalent to assigning the value of one of the two coordinates,
which is, in this case, to be entirely dependent on the other. The
analytical relation which expresses this dependence can be more or less
difficult to discover; but we must obviously always conceive of
its existence, even in cases where our present means would be
insufficient to make it known. It is by this simple
consideration that, independently of the particular verifications on
which this fundamental conception is ordinarily established on
the occasion of such or such line definition, we can demonstrate,
in an entirely general manner, the necessity of the representation.
line analytics by equations.
By taking the same reflections in the opposite direction, we would also
easily demonstrate the geometrical necessity of the representation of
any equation with two variables, in a determined system of
coordinates, by a certain line, of which such a relation would be, in the
absence of no other known property, a
very characteristic definition , and which will have for scientific purpose to fix
the attention immediately on the general progress of the solutions of
the equation, which will thus be noted in the most sensitive
and the simplest way. This painting of equations is one of the advantages
most important fundamentals of analytical geometry, which thereby
reacted to the highest degree on the general improvement of analysis
itself, not only by assigning to purely abstract research
a clearly determined goal and an inexhaustible career, but, under an
even more direct relationship, providing a new
philosophical means of analytical meditation, which could not be replaced
by any other. Indeed, the purely algebraic discussion of an
equation undoubtedly makes it possible to know the solutions of it in the
most precise way, but by considering them only one by one,
so that, by this way, their general progress cannot be conceived.
that as the final result of a long and painful series of
numerical comparisons , after which intellectual activity must ordinarily
be blunted. On the contrary, the geometrical locus of the equation
being only intended to represent distinctly and with
perfect clarity the summary of this set of comparisons, allows it to be
considered directly while completely disregarding the details
which have furnished it, and thereby can to indicate to our mind
general analytical views , which we would hardly have come to
any other way, for lack of a means of clearly characterizing their
object. It is obvious, for example, that the simple inspection of the
logarithmic curve or the curve y = / sin x makes known in a
much more distinct manner the general mode of variations of
logarithms with respect to numbers or of sines with respect to arcs,
than the most careful study could allow. a
logarithm table or a trigonometric table. We know that this process has
today become entirely elementary, and that it is employed
whenever it is a question of clearly grasping the general character of the law
which reigns in a series of precise observations of a any kind.
Returning to the representation of lines by equations, which is our
main object, we see that this representation is, by its
nature, so faithful, that the line could not undergo any
modification, however slight it may be, without determining in
the equation a corresponding change. This complete exactitude
even often gives rise to special difficulties, in that, in our
system of analytical geometry, the simple displacements of the lines being
felt as well in the equations as the
real variations of size or form, one could be exposed. to be
analytically confused with one another, if the surveyors had not
discovered an ingenious method expressly intended to distinguish them
constantly. This method is based on that, although it is
impossible to change analytically at will the position of a line
with respect to the coordinate axes, one can change in
any way the situation of the axes themselves, which is obviously
equivalent; consequently, with the aid of the very simple general formulas by
which this transformation of axes is effected, it becomes easy to
recognize whether two different equations are only the
analytical expression of the same line, variously situated, or relate to each other. to
truly distinct geometrical locations, since, in the first case,
one of them must fit into the other by suitably changing the
axes or the other constants of the coordinate system considered. Of
However, it should be noted on this subject that general inconveniences of
this nature appear, in analytical geometry, to be
strictly inevitable; since the ideas of position being, as we
have seen, the only geometrical ideas immediately reducible to
numerical considerations, and the notions of form being able to be
brought back to them only by seeing in them relations of situation, it is
impossible that analysis does not at first confuse the phenomena of
form with simple phenomena of position, the only ones which the
equations express directly.
To complete the philosophical explanation of the fundamental conception
which serves as a basis for analytical geometry, I believe I should indicate here
a new general consideration, which seems to me particularly
suited to bringing to light this necessary representation of
lines by equations in two variables. It consists in the fact that
not only, as we have established, any definite line must
necessarily give rise to a certain equation between the two
coordinates of any one of its points; but, moreover, any
line definition can be seen as already itself an
equation of that line in a suitable coordinate system.
It is easy to establish this principle, by first making a distinction
preliminary logic relative to the various kinds of definition. The
strictly indispensable condition of any definition is to
distinguish the defined object from any other, by assigning a property
which belongs to it exclusively. But this object can be attained, in
general, in two very different ways: or by a definition
simply _characteristic_, that is to say, indicating a property which,
although really exclusive, does not make known the generation of
the object; or by a really _explicative_ definition, that is to say,
characterizing the object by a property which expresses one of its modes of
generation. For example, considering the circle as the line which,
under the same contour, contains the largest area, we obviously have a
definition of the first kind; while by choosing the property
of having all its points at an equal distance from a fixed point, or any other
similar, we have a definition of the second kind. It is, moreover,
obvious, in general thesis, that even though any object whatsoever would
first be known only by a _characteristic_ definition, it should
not be considered less as susceptible of _explicative_ definitions,
which the further study of this object.
This being said, it is clear that it is not to definitions simply
_caractéristiques_ that the general observation announced can apply.
above, which represents any line definition as
necessarily being an equation of that line in some
coordinate system . We can only hear really
_explicative_ definitions . But, considering only this, the principle is
easy to see. In fact, it is evidently impossible to define the
generation of a line without specifying a certain relation between the
two simple movements, of translation or of rotation, in which
the movement of the point which describes it will be broken down at every moment. Now, by
forming the most general notion of what a _system of
coordinates_ is, and admitting all possible systems, it is clear
that such a relation will be nothing other than the_equation_ of the
proposed line , in a system of coordinates of a nature corresponding to
that of the mode of generation considered. Thus, for example, the
vulgar definition of the circle can obviously be considered as being
immediately the_polar equation_ of this curve, taking for pole
the center of the circle; similarly, the elementary definition of the ellipse or
of the hyperbola, as being the curve generated by a point which moves
in such a way that the sum or the difference of its distances to two
fixed points remains constant, gives on -field, for one or
the other curve, the equation y + x = c, taking for coordinate system
that in which one would determine the position of a point by its
distances to two fixed points, and choosing for these poles the two
given foci; similarly again, the ordinary definition of
any cycloid would provide directly, for this curve,
the equation y = mx, by adopting as coordinates of each point the
greater or lesser arc that it marks on a circle of invariable radius
from from the point of contact of this circle with a fixed line, and the
rectilinear distance from this point of contact to a certain origin taken
on this line. Similar and equally
easy checks can be made with respect to the usual definitions of spirals,
epicycloids, etc. We will constantly find that there exists a certain
system of coordinates, in which we immediately obtain a
very simple equation of the proposed line, by limiting ourselves to writing
algebraically the condition imposed by the mode of generation under
consideration.
Besides its direct importance, as a means of making perfectly
sensible the necessary representation of any line by an equation,
the preceding consideration seems to me to be able to offer a real
scientific utility, by characterizing with exactitude the principal
general difficulty which one meets in the effective establishment of
these equations, and, therefore, providing an indication
interesting in relation to the procedure to be followed in research of this
kind, which, by their nature, cannot include complete
and invariable rules . Indeed, if any definition of line, at
least among those which indicate a mode of generation,
directly provides the equation of this line in a certain system of
coordinates, or to better say constitutes this equation by itself,
it s It follows that the difficulty which is often experienced in discovering
the equation of a curve, according to one or another of its
characteristic properties , a difficulty which is sometimes very great, must
arise essentially only from the condition that one imposes itself
usually to express this curve analytically using a
designated coordinate system, instead of admitting all
possible systems indifferently . These various systems cannot be
regarded, in analytical geometry, as being all equally
suitable; for various reasons, the most important of which will be
discussed below, geometers believe they almost always have to
relate, as much as possible, the curves to the
rectilinear coordinates proper. Now, we can see, from what precedes,
that often these unique coordinates will not be those relative to
which the equation of the curve would be immediately established.
by the proposed definition. The main difficulty in
forming the equation of a line therefore really consists, in
general, in a certain transformation of coordinates. Doubtless,
this consideration does not subject the establishment of these equations to
a true complete general method, the success of which is always
necessarily assured, which, by the very nature of the subject, is
obviously chimerical; but such a view may usefully enlighten us in
this regard on the course which should be adopted in order to achieve the
proposed goal . Thus, after having first formed the preparatory equation which
derives spontaneously from the definition under consideration, it will be necessary, to
obtain the equation relating to the coordinate system which must be
definitively admitted, seek to express as a function of these latter
coordinates those which naturally correspond to the mode of generation in
question. It is on this last work that it is obviously
impossible to give invariable and precise precepts. We can
only say that we will have all the more resources in this regard, that we
will know more of true analytical geometry, that is to say, that we will
know the algebraic expression of a greater number. of
different geometric phenomena .
To complete the philosophical exposition of the design which serves as the
basis of analytical geometry,
considerations relating to the choice of the coordinate system which is, in
general, the most suitable, which will furnish the rational explanation
of the unanimous preference given to the ordinary rectilinear system, a
preference which has hitherto been rather the effect of a feeling empirical
evidence of the superiority of this system, as the exact result of
direct and in-depth analysis .
In order to decide clearly between all the various coordinate systems,
it is essential to distinguish with care the two general points of view
, inverse to each other, specific to analytical geometry,
namely: the relation of the algebra to geometry, based on
representation of lines by equations; and reciprocally the
relation of geometry to algebra based on the painting of
equations by lines.
It is evident that, in any research whatsoever in
general geometry , these two fundamental points of view are necessarily
constantly combined, since it is always a question of passing
alternately, and at intervals, so to speak insensible, from
geometrical considerations to analytical considerations, and from
analytical considerations to geometric considerations. But the
need to separate them here momentarily is no less real;
because the answer to the question of method that we are examining is, in
Indeed, as we are going to see, very far from being able to be the same under
one and the other of these two relations, so that without this
distinction one could not form any clear idea.
Under the first point of view, rigorously isolated, the only reason which
can make one prefer one coordinate system to another, can only be
the greater simplicity of the equation of each line, and the
greater ease of achieving it. . However, it is easy to see that there
does not exist and must not exist any coordinate system deserving in this
respect a constant preference over all the others. Indeed, we
noticed above that, for each proposed geometric definition, we
can conceive of a coordinate system in which the equation of the
line is obtained immediately and is necessarily found to be at the same
time very simple: moreover, this system inevitably varies with the
nature of the characteristic property considered. Thus, the
rectilinear system could not be, in this sense, constantly the most
advantageous, although it is often very favorable; there is
probably not a single one which, in certain particular cases, should not
in this respect be preferred to it, as well as to any other system.
On the contrary, it is not at all the same under the second point of
view. One can, in fact, easily establish, in general thesis, that the
An ordinary rectilinear system must necessarily adapt better than
any other to the painting of equations by the
corresponding geometric places , that is to say, this painting is constantly
simpler and more faithful to it.
Let us consider, for this, that, any coordinate system consisting in
determining a point by the intersection of two lines, the system suitable
for providing the most suitable geometrical locations must be the one
in which these two lines are the simplest possible, this which
initially restricts the choice to being able to relate only to
_rectilinear_ systems . In truth, there is obviously an infinity of systems
which deserve this name, that is to say which use only straight lines
to determine the points, in addition to the ordinary system which assigns as
coordinates the distances to two fixed lines; such would be, for example,
that in which the coordinates of each point would be found to be
the two angles formed by the straight lines which terminate from this point at two
fixed points with the line of junction of the latter; so that
this first consideration is not strictly sufficient to
explain the preference granted unanimously to the ordinary system. But,
on examining in more depth the nature of any
coordinate system , we further recognized that each of the two lines
whose meeting determines the point considered, must necessarily
offer at each instant, among its various unspecified conditions of
determination, a single variable condition, which gives rise to the
corresponding ordinate , and all the other fixed ones, which constitute the _axes_
of the system, taking this term in its broadest mathematical sense
: variation is essential for all positions
to be considered, and fixity is no less so for
there to be means of comparison. Thus, in all
_rectilinear_ systems , each of the two lines will be subject to a
fixed condition , and the ordinate will result from the variable condition. In this respect,
it is obvious, in general thesis, that the system most favorable to the
construction of geometric loci, will necessarily be that according to
which the variable condition of each line will be the simplest
possible, except to complicate for that, if it so. must, the fixed condition.
Now, of all the possible ways of determining two
mobile lines , the easiest to follow geometrically is certainly that
in which, the direction of each line remaining invariable, it only
approaches or moves away more or less from an axis. constant. It
would be, for example, obviously more difficult to clearly imagine
the displacement of a point produced by the intersection of two lines,
which would each rotate around a fixed point making with a
certain axis a greater or lesser angle, as in the
coordinate system previously indicated. This is the true
general explanation of the fundamental property which, by its nature, the
ordinary rectilinear system presents, of being more apt than any other at the
geometric representation of equations, as being the one in which
it is easier to conceive. the displacement of a point as a result
of the change in the value of its coordinates. To clearly feel the full
force of this consideration, it would suffice, for example, to
carefully compare this system with the polar system, in which this
geometric image so simple and so easy to follow, of two straight lines each
moving parallel to the corresponding axis, is replaced
by the complicated picture of an infinite series of concentric circles
cut by a line subject to rotate around a point fixed. It
is, moreover, easy to conceive _a priori_ what must be, for
analytical geometry, the extreme importance of such a
deeply elementary property, which, for this reason, must be reproduced at
every moment and take a progressively increasing value in
all work of this nature [22].
[Note 22: Having to limit myself here to the most
In general, I have not considered several other
elementary disadvantages of less importance, but
nevertheless very serious, presented by the system of
polar coordinates, such as not admitting a
geometric interpretation for the sign of the directing radius,
and even assigning sometimes a single point for various
distinct solutions, from which it follows that the painting of the
equations is necessarily imperfect. Whatever
these drawbacks may be, as several systems other than the
ordinary rectilinear system could also be
exempt from them, they should not be taken into account in establishing the
general superiority of the latter.]
By further specifying the consideration which demonstrates the superiority of
the ordinary coordinate system over any other as regards the painting of
equations, one can even realize the utility of the
usual use in this respect. to take as much as possible the two axes
perpendicular to each other rather than with any other inclination. With
regard to the representation of lines by equations, this
secondary circumstance is no more universally suitable than
we have seen the very nature of the system be; since, depending on the
occasion, any other inclination of the axes may merit the
preference. But, from the opposite point of view, it is easy to see that
rectangular axes constantly make it possible to paint the equations
in a simpler and even more faithful manner. For, with
oblique axes , the space being divided by them into regions whose
identity is no longer perfect, it follows that, if the
geometrical locus of the equation extends simultaneously in all these regions , he
will present there, at the rate of the only inequality of the angles, differences
of figure which, not corresponding to any analytical diversity,
will necessarily alter the rigorous accuracy of the table, by
mingling with the specific results of algebraic comparisons. For example,
an equation like x ^ m + y ^ m = c, which, by its perfect symmetry, should
obviously give a curve composed of four identical quarters, will be
represented, on the contrary, by taking non-rectangular axes, by
a geometrical locus of which the four parts will be unequal. We see that
the only way to avoid any discrepancy of this kind is to assume
the angle of the two axes to be right.
The preceding discussion makes it clear that, if, under one of the two
fundamental points of view continually combined in
analytical geometry , the rectilinear coordinate system proper has
no constant superiority over any other; as it is not either
in this constantly inferior respect, his greater necessary and
absolute aptitude for painting equations must generally cause him to be given
preference, although it may evidently happen, in some
particular cases , that the need to simplify the equations and
obtain them more easily determines surveyors to adopt a less
perfect system . It is, in fact, according to the rectilinear system, that
the most essential theories of
general geometry are ordinarily constructed , intended to express analytically
the most important geometrical phenomena . When we consider it necessary to choose
another, it is almost always the polar system that we stop at,
this system being of a nature quite opposite to that of the
rectilinear system so that the too complicated equations relative to
this one become, in general, sufficiently simple compared to
the other. Polar coordinates, moreover, often have the advantage of
having a more direct and natural concrete meaning,
as happens in mechanics for the geometrical questions to which
the theory of rotational motions gives rise, and in almost all
cases of celestial geometry. .
In order to simplify the exposition, we have so far considered the
fundamental conception of analytical geometry only in relation to
only plane curves, the general study of which had been the sole object of
the great philosophical renovation carried out by Descartes. It is
now a question , to complete this important explanation, to show
briefly how this elementary thought was extended,
about a century later, by our illustrious Clairaut, to the general study
of surfaces and curves with double curvature. The considerations
indicated above will allow me to confine myself on this subject to a
rapid examination of what is strictly specific to this new case.
The entire analytical determination of a point in space
obviously requires assigning the values of three coordinates; for example,
according to the system most frequently adopted and which corresponds to
the _rectilinear_ system of plane geometry, the distances from this point to
three fixed planes, usually perpendicular to each other, which
presents the point as the intersection of three planes whose direction
is invariable. One could also use the distances of the
moving point to three fixed points, which would determine it by the meeting of
three spheres with constant center. Likewise, the position of a point would be
defined by giving its greater or lesser distance to a fixed point, and
the direction of this distance, by means of the two angles made by this
straight line with two invariable axes; it is the _polar_ system specific to
three-dimensional geometry; the point is then constructed by
the intersection of a sphere with constant center with two right cones with
circular base whose axes and the common vertex do not change. In a
nutshell, there is evidently in this case at least the same infinite variety
between the various possible coordinate systems that we have already
observed for two-dimensional geometry. In general, one should
conceive of a point as always determined by the intersection of
any three surfaces, as it was previously by that of two
lines; each of these three surfaces has similarly all its
constant conditions of determination, except one, which gives rise to the
corresponding coordinate, whose specific geometric influence is thus
to constrain the point to be located on this surface.
That being said, it is clear that if the three coordinates of a point are
entirely independent of each other, this point will be able to take
successively in space all the possible positions. But, if the
point is subject to remaining on a certain surface, defined in
any way, then two coordinates are obviously sufficient to
determine its situation at each moment, since the proposed surface will
take the place of the condition imposed by the third coordinate. We
must therefore necessarily conceive in this case, from the point of view
analytic, the latter coordinate as a determined function of the
two others, the latter remaining between them completely independent.
Thus, there will be between the three variable coordinates a certain
permanent equation, and which will be unique in order to correspond to the
precise degree of indeterminacy of the position of the point. This equation, more
or less easy to discover, but always possible, will be the
analytical definition of the proposed surface, since it will have to be verified for
all the points of this surface, and only for them. If the surface
undergoes
any change, even a simple displacement, the equation will have to undergo a corresponding modification more or less
deep. In a word, all the geometric phenomena whatsoever
relating to surfaces will be capable of being translated by certain
equivalent analytical conditions specific to equations with three
variables, and it is in the establishment and interpretation of this
general and necessary harmony that will consist essentially the
science of three-dimensional analytical geometry.
Considering then this fundamental conception from the
opposite point of view , we see in the same way that any equation with three variables
can be, in general, represented geometrically by a
determined surface , originally defined according to the property
very characteristic, that the coordinates of all its points
always preserve between them the relation stated in this equation. This
geometric locus will obviously change, for the same equation, according to the
coordinate system which will be used to construct this table. By
adopting, for example, the rectilinear system, it is clear that in
the equation between the three variables x, y, z, each particular value
assigned to z, will give an equation between x and y, whose
geometric locus will be a certain line located in a plane parallel to
the x, y plane, and at a distance from the latter equal to the value of z,
so that the total geometrical locus will appear as compound
of an infinite series of lines superimposed in a series of
parallel planes , except for the interruptions which may exist, and will
therefore form a real surface. It would be the same considering
any other coordinate system, although the geometric construction
of the equation became more difficult to follow.
Such is the elementary conception, complement of the mother idea of
Descartes, on which is founded the general geometry relative
to surfaces. It would be useless to take again directly here the other
considerations indicated above in relation to the lines, and which each one
can easily extend to the surfaces, either to show that any
definition of a surface by any mode of generation is
really a direct equation of this surface in a certain
coordinate system , that is to say to determine between all the various
possible coordinate systems which is generally the most suitable.
I will only add, in this latter respect, that the
necessary superiority of the ordinary rectilinear system, as regards the painting of
equations, is evidently even more pronounced in
three-dimensional analytical geometry than in that of two, because of the
geometric complication. incomparably greater which would
then result from the choice of any other system, as can be seen from the
most noticeable by considering, by contrast, the
polar system in particular, which is, for surfaces as for
curves, and by virtue of the same patterns, the most used after the
rectilinear system proper.
In order to complete the general exposition of the fundamental conception
relative to the analytical study of surfaces, we shall have yet to
examine philosophically, in the fourteenth lesson, a final
improvement of the greatest importance, which Monge recently
introduced into the elements themselves. of this theory, for the
classification of surfaces into natural families, established according to the
mode of generation, and expressed algebraically by equations
common differentials, or by finite equations containing
arbitrary functions.
Let us now consider the last elementary point of view of
analytical three-dimensional geometry, that which relates to the
algebraic representation of curves, considered in space in
the most general way. By continuing to follow the principle constantly
employed above, that of the degree of indeterminacy of the geometrical locus,
corresponding to the degree of independence of the variables, it is obvious, in
general thesis, that, when a point must be located on a certain
any curve, a single coordinate is enough to complete the complete
determination of its position, by the intersection of this curve
with the surface resulting from this coordinate. Thus, in this case, the
other two coordinates of the point must be conceived as
functions necessarily determined and distinct from the first. As
a result, any line seen in space, is represented
analytically, not by a single equation, but by the system of
two equations between three coordinates of any of its
points. It is clear, in fact, on the other hand, that each of these
equations, considered separately, expressing a certain surface, their
set presents the line proposed as the intersection of two
determined surfaces. This is the most general way of conceiving
the algebraic representation of a line in
three-dimensional analytical geometry . This conception is ordinarily considered
too narrowly, when one confines oneself to considering a line as
determined by the system of its two _projections_ on two of the
coordinated planes , a system characterized analytically by this peculiarity
that each of the two equations of the line then only contains two
of the three coordinates, instead of simultaneously containing the three
variables. This consideration, which consists in looking at the line as
the intersection of two cylindrical surfaces parallel to two of the three
coordinate axes, in addition to the disadvantage of being limited to the system
rectilinear rectilinear, has the defect, when we believe we have to be reduced
strictly to it, of introducing unnecessary difficulties in the
analytical representation of the lines, since the combination of these
two cylinders could obviously not always be the most suitable
for forming the equations of a line. Thus, considering this
fundamental notion in its entire generality, it will be necessary, in each case,
from among the infinity of pairs of surfaces whose intersection could
produce the proposed curve, to choose the one which lends itself best to
the establishment of the equations, as consisting of the best
known surfaces . For example, is it a question of analytically expressing a circle in
space, it will obviously be preferable to consider it as
the intersection of a sphere and a plane, rather than following any other
combination of surfaces which could also produce it.
In truth, this way of conceiving of the representation of lines by
equations in analytical three-dimensional geometry, generates,
by its nature, a necessary disadvantage, that of a certain
analytical confusion, consisting in that the same line can be
thus find expressed, with the same coordinate system, by an infinity of
pairs of different equations, given the infinity of pairs of surfaces which
can form it, which may present some difficulties for
recognize this line through all the algebraic disguises of which
it is susceptible. But there is a very simple general process
for eliminating this drawback, for depriving oneself of the facilities which
result from this variety of geometric constructions. It suffices, in
fact, whatever the analytical system originally established for a
certain line, to be able to deduce therefrom the system corresponding to a
single pair of uniformly generated surfaces, for example, to that
of the two cylindrical surfaces which _project_ the proposed line on
two of the coordinate planes, surfaces which obviously will always be
identical in whatever way the line has been obtained, and not
will vary only when that line itself changes. Now, by choosing
this fixed system, which is effectively the simplest, we can
generally deduce from the primitive equations those which
correspond to them in this special construction, by transforming them, by
two successive eliminations, into two equations each containing
only two of the coordinates variables, and which by that alone will suit
the two projection surfaces. Such is really the principal
destination of this sort of geometrical combination, which
thus offers us an invariable and certain means of recognizing the identity of the
lines in spite of the sometimes very great diversity of their equations.
After having considered as a whole the fundamental conception of
analytical geometry under the main elementary aspects that it
can present, it is appropriate, to complete, from a
philosophical point of view, such a sketch, to point out here the
general imperfections which this conception still presents. , either relatively to
geometry, or relatively to analysis.
With regard to geometry, it should be noted that the equations are up to
now only suitable for representing whole geometric loci, and
not at all determined portions of these geometrical loci. It would
however be necessary, in several circumstances, to be able to express
analytically a part of a line or a surface, and even a
_discontinuous_ line or surface composed of a series of sections belonging to
distinct geometric figures, for example the contour of a polygon or
the surface of a polyhedron. Thermology especially frequently gives rise
to similar considerations, to which our
present analytical geometry is necessarily inapplicable. Nevertheless, it is important
to observe that, in recent times, the work of M. Fourier on
discontinuous functions has begun to fill this great gap,
and thereby directly introduced a new
essential improvement in the fundamental conception of Descartes. But this
way of representing heterogeneous or partial forms, being
based on the use of trigonometric series proceeding according to the sines
of an infinite series of multiple arcs, or on the use of certain
definite integrals equivalent to these series and of which the integral
general is ignored, still presents too many complications to
be able to be immediately introduced into the proper system of
analytical geometry .
With regard to analysis, we must begin by recognizing that
the impossibility in which we are to design geometrically for
equations containing four, five variables or a greater number, a
representation analogous to those included in all the equations with
two or three variables, should not be considered as an
imperfection of our system of analytical geometry, because it
obviously stems from the very nature of the subject. Analysis being necessarily
more general than geometry, since it relates to all
possible phenomena, it would be unphilosophical to constantly want to
find among the only geometrical phenomena a
concrete representation of all the laws that analysis can express. But there is
another lesser imperfection which must really
be seen to arise from the very way we conceive of
analytical geometry. It consists in that our representation
current equations with two or three variables by lines or
surfaces is obviously always more or less incomplete, since, in
the construction of the geometrical locus, we only consider the
_real_ solutions of the equations, without taking any account of the
imaginary solutions . The general course of the latter would however, by
its nature, be just as susceptible as that of the others of a
geometric painting . It results from this omission that the graphic table of
the equation is constantly imperfect, and sometimes even to the point that
there is no more geometric representation, when the equation admits only
imaginary solutions. However, even in the latter case, there
it would obviously be appropriate to distinguish geometrically
equations as different in themselves as these, for example,
/ [x ^ 2 + y ^ 2 + 1 = 0, /; x ^ 6 + y ^ 4 + 1 = 0, /; y ^ 2 + e ^ x = 0. /] We also know that this
main imperfection often involves, in analytical geometry in
two or three dimensions, a host of secondary disadvantages, due
to the fact that several analytical modifications are found not to correspond
to any geometrical phenomenon.
One of our greatest present-day geometers, M. Poinsot, has presented a
very ingenious and very simple consideration, to which not
enough attention has generally been paid, and which allows, when the equations are
uncomplicated, to conceive the graphic representation of
imaginary solutions , limiting oneself to painting their relations when they are
real [23]. But this consideration, which it would be easy to generalize
abstractly, is so far too unlikely to be effectively
employed, because of the extreme state of imperfection in which
the algebraic resolution of equations still lies , and hence it results either that the
form of the imaginary roots is most often ignored, or that it
presents too great a complication; so that new work
is essential in this regard, before we can regard as
filled this essential gap in our analytical geometry.
[Note 23: M. Poinsot showed, for example, in his
excellent _memoire on the analysis of angular sections_,
that the equation x ^ 2 + y ^ 2 + a ^ 2 = 0, usually discarded as
not having of geometric locus, can be represented, in
the simplest and most clear way, by an
equilateral hyperbola, which fulfills the same function with regard to it
as the circle for the equation x ^ 2 + y ^ 2-a ^ 2 = 0.]
The philosophical exposition tried in this lesson of the
fundamental conception of analytical geometry, clearly shows us that
this science consists essentially in determining what is, in
general, the analytical expression of this or that geometric phenomenon
proper to lines or surfaces, and conversely to discovering
the geometric interpretation of such or such analytical consideration.
We now have to examine, by confining ourselves to
the most important general questions , how the geometers succeeded in
effectively establishing this beautiful harmony, and thus in imparting to
geometrical science, considered as a whole, the
perfect character of rationality and of simplicity that it presents today so
eminently. Such will be the essential object of the two following lessons, one
devoted to the general study of lines, and the other to the general study
of surfaces.
THIRTEENTH LESSON.
SUMMARY. Two-dimensional _general_ geometry.
According to the course usually adopted up to this day for
the exhibition of geometrical science, the really
essential destination of analytical geometry has only been felt in a
very imperfect way, which in no way corresponds to the opinion that
the true geometers form of it, since the extension of the
analytical conceptions to rational mechanics made it possible to rise
to some general ideas on mathematical philosophy. The
fundamental revolution brought about by the great thought of Descartes is
not yet worthily appreciated in our mathematical education, even
the highest. In the manner in which it is ordinarily presented and
especially employed, this admirable method would initially seem to have
no other real aim than to simplify the study of conical sections, or of
some other curves, always considered one by one following the spirit
of ancient geometry, which would doubtless be of very little
importance. We have not yet properly felt that the real
distinctive character of our modern geometry, which constitutes its
incontestable superiority, consists in studying, in an entirely
general manner , the various questions relating to any lines or
surfaces, in transforming the considerations and
geometric research into considerations and analytical research.
It is remarkable that in the establishments, even the most justly
famous, devoted to high mathematical instruction, no
truly dogmatic course in general geometry has been instituted, conceived in a
manner both distinct and complete [24]. However, such a study
is the most suitable for clearly manifesting the true
philosophical character of mathematical science, by demonstrating with
perfect clarity the general organization of the relation from the abstract to the concrete
in the mathematical theory of any order of phenomena.
natural.
[Note 24: The profound mediocrity that we observe
generally in this respect, especially in the teaching of the
elementary part of mathematics, although two centuries
have already passed since the publication
of Descartes' geometry , shows how our
ordinary mathematical education is still far from corresponding to the true state
of science ; which is no doubt, in large part, and we
should not conceal it, from the extreme inferiority of
most people to whom such an
important teaching is entrusted , on whose top management the
real leaders of science are not due. otherwise admitted to
exercise no regular and permanent influence.]
These considerations sufficiently indicate what may be, besides its extreme
philosophical importance, the special and direct utility of the exposition
to which the outline of this work now leads us. It is
therefore a question , starting from the fundamental conception explained in the
previous lesson , relative to the analytical representation of
geometric forms , to examine how geometers have managed to reduce
all questions of general geometry to pure questions
of analysis. , by determining the analytical laws of all
geometrical phenomena , that is to say the algebraic modifications which
correspond to them in the equations of lines and surfaces. I do not
I will deal first of all with curves, and even with plane curves,
reserving for the next lesson the general study of surfaces and
curves with double curvature. The spirit of this work prescribes moreover
to confine oneself to the philosophical examination of the most
important general questions , and above all to rule out any application to
particular forms . The essential aim which we must have in view here is
only to observe with precision how the
fundamental conception of Descartes established the general system of
geometrical science on rational and definitive bases. Any other study
would fit into a special treatise on geometry; but, as for this one,
it is essential for the object we are proposing. One can
undoubtedly conceive _a priori_, as I indicated in the
preceding lesson , that, once the subject of geometrical researches
represented analytically, all the _accidens_ or any phenomena of
which it is susceptible must necessarily include a
similar interpretation. But it is clear that such a consideration in
no way dispenses, even under the simple philosophical relation,
from studying the effective organization of this general harmony between
geometry and analysis, of which we would otherwise only form an idea.
vague and confused, entirely insufficient.
The first and the simplest question which one can propose
relative to any curve, is to know, from its
equation [25], the number of points necessary for its determination. Besides
the proper importance of such a notion, which has not yet been established
in a sufficiently rational manner, I believe I should set out with some
development the general solution of this elementary problem, because
it seems to me eminently suitable. , with regard to the method, given
the extreme simplicity of the corresponding analytical considerations, to
capture the true spirit of analytical geometry,
that is to say the necessary and continuous correlation between the point of view
concrete and abstract point of view.
[Note 25: I will always consider, for the sake of
clarity, unless formal warning, the
ordinary rectilinear coordinate system , either in this lesson
or in the next.]
To fully resolve this question, we must distinguish two cases,
depending on whether the proposed curve is defined analytically by its
most general equation, that is to say, suitable for all the positions
of the curve relative to the axes, or by a particular and
simpler equation , which takes place only in a certain situation of the curve with
respect to the axes.
In the first case, it is evident that the condition, on the part of the
curve, of having to pass through a given point, is analytically equivalent to the
fact that the arbitrary constants contained in its general equation
retain between them the relation marked by the substitution of the
particular coordinates of this point in this equation. Each
given point thus imposing on these constants a certain algebraic condition,
for the curve to be entirely determined it will therefore be necessary to assign
a number of points equal to the number of arbitrary constants contained
in its equation. This is the general rule. It should however be
noted that it could be misleading, and indicate
too large a number of points, if, in the proposed equation, the number of
distinct terms containing the arbitrary constants was less than
that of these constants, in which case one would obviously have to judge the
number of points necessary for the entire determination of the curve,
only by that of these terms, which would geometrically mean
that the constants considered could then to experience certain
changes without any result for the curve. Such would be, for
example, the case of the circle, if we defined it as the curve described
by the vertex of an angle of invariable magnitude which moves in such a way
that each of its sides always passes through a certain fixed point. It
is therefore necessary, for more generality, to count separately the number of
constants entering the equation of the proposed curve and the number
of terms that contain them, and determining how many points requires
the entire specification of the curve by the smaller of these two
numbers, unless they are equal.
When a curve is initially defined only by an equation of the
kind we named _particular_ above, we
can, using an invariable and very simple transformation, make
this case fit into the previous one, by _generalizing_ suitably
the proposed equation. It suffices, for this, to relate the curve,
according to known formulas, to a new system of axes, whose
situation in relation to the former is regarded as indeterminate. If
this transformation does not essentially change the
analytical composition of the primitive equation, it will be the proof that it was
already sufficiently general; otherwise, it will have become so,
and the question will therefore be easily resolved by applying the
rule previously established. We can even observe, to simplify
this solution even further, that this generalization of the equation will
always introduce, whatever the primitive equation, three
new arbitrary constants, namely the two coordinates of the
new origin and the inclination of the new axes. on the elders; in
so that, without carrying out the calculation, we will be able to know the number of
arbitrary constants which would enter into the most general equation,
and consequently directly deduce from it the number of points necessary for the
determination of the proposed curve, at least every time that it
will be possible to be certain in advance, which takes place very frequently, that the
number of terms which would contain these constants would not be
less than that of the constants themselves.
In order to show how easily the
general solution of this question can be reached , it is important to note that, the
analytical operation prescribed to solve it is reduced to a simple
enumeration, this enumeration can be done even before the equation
of the curve is obtained, and only according to its geometric definition.
It suffices, in fact, to analyze this definition from this point of view,
by estimating how many given points, or lines given either in
length, or in direction, or of given circles, etc., it requires for
the entire determination of the proposed curve. That being said, we will also know
in advance how many arbitrary constants he will have to enter into
the most general equation of this curve, considering that each
fixed point given by the definition will introduce two, each line
given also two, each length given one, each circle
fully given three, etc. We can therefore immediately judge by this
the number of points required for the determination of the curve, with as much
accuracy as if we had before our eyes its general equation; with
this close however to the restriction indicated above for the case where
the number of terms containing the arbitrary constants would be
lower than that of the constants; restriction that can often be
recognized as inapplicable, if the analysis of the proposed definition has
clearly shown that the data it prescribes could
not vary in any way, either individually or together, without resulting in a change
for the curve any. But when this restriction
must be really applied, this consideration will first provide
only an upper limit of the number sought, which can then be
fully known only by actually consulting the general equation.
I have hitherto supposed that the points by which one wishes to determine the
course of a line were absolutely arbitrary; but, to complete
the method, it is necessary to examine the case where one would introduce among them
_singular_ points, that is to say distinct from all the others by
any characteristic property, such as what we call
_foyers_ in conical sections, _ vertices_, _centres_,
points of_inflection_ or _bending_, etc. These points all having
for the character of being unique, or at least determined, in the same
curve, their two coordinates are therefore each a
determined function , known or unknown, constants which
exactly specify the proposed curve. Thus, to give only one of these points,
is to impose on these arbitrary constants two algebraic conditions,
which, consequently, is analytically equivalent to giving two
ordinary points . The general and very simple rule is therefore reduced, in this
respect, to always counting each _singular_ point as two, by
whatever property it may be defined: apart from that, we shall return
to the law established above.
Any special application of the general theory that I come from
to indicate here would be inappropriate. I believe, however, useful to note,
concerning this application, that the number of points necessary for
the entire determination of each curve, although constituting a
very important circumstance, is not as intimately related as one
would initially believe. , either to the analytical nature of the equation, or to the
geometric shape of the line. Thus, for example, we find, according to the
previous method, that the ordinary parabola, and even parabolas of
all degrees, the logarithmic, the cycloid, the Archimedean spiral,
etc., also require four points for their determination, although we
could not discover until
curves as different from the analytical point of view as from the
geometrical point of view. However, it is likely that this analogy
should not be entirely isolated.
I will choose, as a second interesting example, among the
elementary questions relating to the general study of lines, the determination
of the _centers_ in any plane curve. The geometric character
of the _centre_ of a figure being, in general, to be the midpoint of all
the strings which pass through it, it obviously follows that, if we place there
the origin of the rectilinear coordinate system, the points of the
figure will have, two by two, with respect to such an origin,
equal coordinates and opposite sign. We can therefore
immediately recognize , from the equation of any curve, whether its
center has the actual origin of the coordinates, since it suffices
to examine whether this equation is not altered, by changing it to the
time the signs of the two variable coordinates, which requires, in
case there is between that of algebraic, rational and
entire, all the terms are of even degree or any degree of
odd, depending on the degree of 'equation. That said, when such a
change disturbs the equation, we must move the origin in an
indeterminate manner , and seek to dispose of the two arbitrary constants that
this transformation introduces into the equation for the coordinates of
the new origin, so that the equation can enjoy,
relative to the new axes, the previous property. If, by
suitable real values of the coordinates of the new origin, we
can eliminate all the terms which prevented the equation from
presenting this analytical character, the curve will have a center whose
position will be known by these values: otherwise, it will be
noted that the curve has no center.
Among the questions of general two-dimensional geometry whose
complete solution depends only on ordinary analysis, I think I should
again indicate here that which relates to the determination of the
conditions of the _similitude_ between any curves of the same
_genre_, that is to say susceptible of the same definition or _equation_,
which only distinguishes them from one another by the various values of
certain arbitrary constants relating to the magnitude of each
of them. This question, important in itself, is all the more
interesting from the point of view of the method, since the geometrical phenomenon which
it is then a question of characterizing analytically, is obviously
purely relative to the form, and in no way a situation phenomenon,
which, as we noticed in the previous lesson, gives
always give rise to special difficulties in relation to our system
of analytical geometry, where the ideas of position are alone
directly considered.
The use of differential analysis would immediately provide the
solution to this general problem, extending to curves, as
appropriate, the elementary definition of similitude for
rectilinear figures . It would suffice, in fact, 1º to calculate, from the equation
of each of the two curves, the angle of _contingence_ at
any point , and to express that this angle has the same value in the two
curves for points correspondents; 2º according to the
general differential expression of the length of
each curve, to express that the homologous elements of the two curves
are between them in a constant relation. The analytical conditions of
similarity would thus be found to depend on the first two
functions derived from the ordinate referred to the abscissa. But the problem
can be solved in a much simpler way, and nevertheless just
as general, though less direct, by the simple use of
ordinary analysis .
For this, we must first notice an elementary property that
can always present two similar figures of any shape,
when they are placed in a _parallel_ situation, that is to say,
parallel to the homologous elements of the other, which the similitude
obviously allows us to do constantly. In this situation, it is
easy to see that, if we join two by two by straight lines the
homologous points of the two figures, all these junction lines will
necessarily concur at a single point, from which their lengths,
counted up to l 'one and the other of the two similar figures will have
a constant relationship between them, equal to that of the two figures. It
follows immediately from this property, considered from the
analytical point of view, that, if the origin of the rectilinear coordinates is
supposed to be placed at the particular point of which we have just spoken, the
homologous points of two similar curves will have
constantly proportional coordinates , so that the equation of the first
curve will have to fit into that of the second, by changing y x to mx,
and y to my, m being an arbitrary constant equal to the ratio linear
of the two figures. With polar coordinates z and / varphi, whose
pole would be placed at the same point, the two equations would become
identical by changing only z to mz in one of them, without
changing / varphi. The verification of such an algebraic character will
therefore obviously suffice to note the similarity. But, from its
non-verification, it is clear that one should not conclude
immediately the dissimilarity of the two compared curves, since
the origin or the pole could not be placed at the single point for
which this relation takes place, or even that the two curves could
not be posed currently in the _parallel_ situation. It is
nevertheless easy to generalize and complete the method under both
of these two ratios, although it initially seems
analytically impossible to modify the relative situation of two curves. It
will suffice to change, using known formulas, both
the origin and the direction of the axes if the coordinates are rectilinear,
or the pole and the direction of the axis if they are polar, but in
performing this transformation only in one of the two equations.
We will then seek to have the three arbitrary constants
introduced by this, so that this equation thus modified presents,
relative to the other, the analytical property indicated. If this
relation can take place according to some real values of the
arbitrary constants, the two curves will be similar; otherwise, their
dissimilarity will be noted.
Although it is inappropriate to consider here any
special application of the preceding theory, I nevertheless believe it useful to indicate
on this subject a general remark. It consists in that,
whenever the equation of a curve,
arrangement of the axes, will contain only one arbitrary constant,
all the curves of this kind will necessarily be similar to each
other. We can increase the utility of this observation, in that, without
even considering the equation of the curve, it will suffice to examine, in this
case, whether its primitive geometrical definition makes depend only on a
single datum the full determination of its greatness [26]. When, on the
contrary, the simplest equation of the proposed curve contains
two or more arbitrary constants, or, what is exactly
equivalent, when the definition will make its magnitude depend on several
distinct data, curves of this kind cannot be alike
only with the help of certain relations between these constants or these data,
which will ordinarily consist in their proportionality. It is thus
that all the parabolas of the same degree, moreover any, are
similar to each other, as well as all the logarithmics, all
the ordinary cycloids, all the circles, etc .; while two
ellipses or two hyperbolas, for example, are only alike insofar
as their axes are proportional.
[Note 26: This property, which is an
obvious consequence of the theory indicated above, could
moreover be established directly by a
very simple consideration . It would suffice to notice that, in this case, the
various curves of this kind might coincide by
constructing them on a different scale, from which
clearly results their necessary similarity.]
I confine myself to this small number of general questions relating to
lines, among those whose complete solution depends only on
the ordinary analysis. One should not include the determination of what
are called the _focuses_, the search for the _diameters_, etc., and
several other problems of this kind, which, although capable
of being proposed and solved for any curves, have no
real interest except in conical sections. With respect to
_diameters_, for example, c '
mediums of any system of parallel strings, it is easy to
form a general method for deducing from the equation of a curve
the common equation of all its diameters. But such a consideration
cannot facilitate the study of a curve unless the diameters are
found to be simpler and better known lines than the
original curve ; and even this research is only really useful
when all the diameters are straight lines. However, this is what only takes
place in quadratic curves. For all the others, the
diameters are, in general, curves as little known and often
even more difficult to study than the curve proposed. That is why
I must not here consider such a question, nor any other
similar, although, in the special treatises on analytical geometry,
it is moreover convenient to present them first, as far as possible,
from an entirely general point of view.
I therefore immediately proceed to the examination of the theories of
general two-dimensional geometry which can only be fully established
with the aid of transcendent analysis.
The first and simplest of them consists in
determining tangents to plane curves. Having had occasion, in
the sixth lesson, to indicate the general solution of this important
problem, according to each of the various fundamental points of view peculiar to
transcendent analysis, it is unnecessary to return to it here. I will
only observe on this subject that the fundamental question thus
considered supposes known the point of contact of the line with the
curve, while the tangent can be determined by several other
conditions, which must then be included in the previous one, by
determining beforehand the co-ordinates of the point of contact, which
is usually very easy. Thus, for example, if the tangent is
subject to passing through a given point outside the curve, the
coordinates of this point must satisfy the general formula of
the equation of the tangent to this curve, which formula contains the
unknown coordinates of the point of contact, this last point will be
determined by such a relation combined with the equation of the
proposed curve . Likewise, if the sought tangent must be parallel to a
given line, it will be necessary to equal the general coefficient which marks its
direction according to the coordinates of the point of contact with that which
determines that of the given line, and the combination of this
condition with the equation of the curve will again reveal these
coordinates.
In order to consider the problems
relating to tangents from a more extended point of view , it may be useful to express clearly the
relation which must exist between the two arbitrary constants
contained in the general equation of a straight line and the various
constants specific to any given curve, so that the line
is tangent to the curve. For this purpose, it suffices to observe that the
two constants by which the
position of the tangent is fixed at each moment being known functions of the coordinates of the
point of contact, the elimination of these two coordinates between these two
formulas and l The equation of the proposed curve will provide a relation
independent of the point of contact and containing only the constants
of the two lines, which will be the analytical character sought of the phenomenon
of an indeterminate contact. We would use, for example, such
expressions to determine a common tangent to two given curves,
by calculating the two constants specific to this line according to the two
relations which would thus involve its contact with one and the other
curve.
The fundamental question of tangents is the starting point for
several other more or less important general research
relating to curves, which it is easy to make depend on. The most
direct and the simplest of these secondary questions consists in the
determination of the rectilinear _asymptotes_, or at least of the _asymptotes_
, the only ones, in general, which it is interesting to
know, because they alone really contribute to facilitate
the study of a curve. We know that the_asymptote_ is a straight line which
approaches indefinitely and as close as we want to a curve, without
however ever being able to reach it rigorously. It can therefore be
considered as a tangent whose point of contact moves away to
infinity. Thus, to determine it, it suffices to assume infinite the
coordinates of the point of contact in the two general formulas which
express, according to the equation of the curve, as a function of these
coordinates, the two constants by which the position is fixed. of
the tangent. If these two constants then take real
and mutually compatible values , the given curve will have asymptotes of which one
such calculation will make known the number and the situation; if these values are
imaginary or incompatible, this will be the proof that the proposed curve
has no asymptotes, at least rectilinear. We see that this
determination is exactly analogous to that of a tangent led by
a point of the curve whose coordinates are finite. It will
only happen , in a rather large number of cases, that the two
sought values will present themselves in an indeterminate form, which is a
general drawback of algebraic formulas, although it must
undoubtedly take place more frequently by attributing
values to the variables. infinite. But we know that there is an analytical method
general to estimate the true value of any similar expression; it
will therefore suffice to resort to it.
We can also relate, although in a much less direct manner, to
the theory of tangents, the whole theory of the various
_singular_ points , the determination of which contributes eminently to the
knowledge of any curve which presents them, such as the points
of_inflection_ , _multiple_ points, _ cusp_ points,
etc. With respect to the points of_inflexion_, for example, that is to say to
those where a curve of concave becomes convex, or of concave convex, we
must first examine the analytical character immediately proper to the
concavity or convexity, which depends on how the
direction of the tangent varies . When the curve is concave towards the axis of the
abscissa, it forms an angle with it smaller and smaller as
it moves away from it; on the contrary, when it is convex, the angle which
it makes with the axis becomes larger and larger as it moves
further away from it. We can therefore directly recognize, from the equation
of a curve, the direction of its curvature at each moment: it suffices
to examine whether the coefficient which marks the inclination of the tangent,
that is to say the function derived from the ordinate, takes
increasing values or decreasing values as the ordinate
increases; in the first case, the curve turns its convexity towards the
abscissa axis ; in the second, its concavity. That said, if there is
_inflexion_ at some point, that is to say if the curvature changes
direction, it is clear that at this point the inclination of the tangent will have
become a _maximum_ or a _minimum_, depending on whether it is the passage
from convexity to concavity, or the reverse passage. We will therefore find
at what points this phenomenon can take place, with the help of the
ordinary theory of _maxima_ and _minima_, whose application to this research
will obviously show that, for the abscissa of the inflection point, the
second derivative function of the proposed ordinate must be zero, which
will suffice to determine the existence and position of this point. This
research can thus be related to the theory of tangents,
although it is usually presented according to the theory of the
osculating circle . It would be the same, with more or less difficulty,
relatively to all the other _singular_ points.
A second fundamental problem presented by the general study of
curves, and the complete solution of which requires a more extensive use of
transcendent analysis, is the important question of the measurement of the
_curvature_ of the curves by means of the _osculator_ circle at each point. ,
the discovery of which alone would be enough to immortalize the name of the great
Huyghens.
The circle being the only curve which presents in all its points a
uniform curvature, all the greater moreover as the radius is
smaller, when the surveyors proposed to submit to a
precise estimate the curvature of any other curve whatever. , they
naturally had to compare it at each point with the circle which could have
with it the most intimate contact possible, and which they named, for
this reason, circle _osculateur_, in order to distinguish it from circles
simply _tangens_, which are in number infinite at the same point of curve,
while the osculating circle is obviously unique. Considering
this question from another aspect, it is understood that the curvature of a
curve in each point could also be estimated by the greater or
lesser angle of two consecutive elements, which one calls angle of
_contingence_. But, it is easy to recognize that these two measurements
are necessarily equivalent, since the center of the osculating circle
will be all the more distant as this angle of contingency will be more obtuse:
we even see, from the analytical point of view, that the expression of the radius
of this circle immediately gives the value of this angle. From this
obvious conformity of the two points of view, the surveyors must have
usually preferred the consideration of the osculating circle, as
more extended and lending itself better to the deduction of other theories.
geometric patterns that relate to this fundamental concept.
This being said, the simplest and most direct way of determining
the osculating circle consists in considering it, according to the
infinitesimal method proper, as passing through three points
infinitely close to the proposed curve, or, in other words. terms, as
having with it two common consecutive elements, which
clearly distinguishes it from all simply tangent circles, with which the
curve has only one common element. It follows from this notion,
having regard to the construction necessary to describe a circle passing
through three given points, that the center of the osculating circle, or what we
called the _centre of curvature_ of the curve at each point, can be
regarded as the point of intersection of two infinitely
close normals , so that the question is reduced to finding this last point.
Now, this research is easy, by forming, according to the general equation
of the tangent to any curve, that of the normal which is
perpendicular to it, and then making it vary by an infinitely
small quantity , in this last equation, the coordinates of the point of
contact, in order to pass to the infinitely close normal: the
determination of the solution common to these two equations, which are of the
first degree with respect to the two coordinates of the point of intersection,
is enough to find the two general formulas which express the
coordinates of the center of curvature of a curve at any point.
Once these formulas have been obtained, the search for the radius of curvature
no longer offers any difficulty, since it is reduced to calculating the
distance from this center of curvature to the corresponding point of the curve.
By calling / alpha, / beta, the rectilinear coordinates of the center of
curvature of any curve at a point whose coordinates are
x, y, and naming the radius of the curve r, we find the
known formulas by this method . / [/ alpha =
x- / frac {/ frac {dy} {dx} / left (1 + / frac {dy ^ 2} {dx ^ 2} / right)} {/ frac {d ^ 2y} {dx ^ 2}}
,; / beta
= y + / frac {/ left (1 + / frac {dy ^ 2} {dx ^ 2} / right)} {/ frac {d ^ 2y} {dx ^ 2}}, /]
/ [r = / frac { / left (1 + / frac {dy ^ 2} {dx ^ 2} / right) ^ {/ frac {3} {2}}} {/ frac {d ^ 2y}
{dx ^ 2}} /]
We can imagine of what importance is the determination of the radius of
curvature, and how much the discussion of the general manner in which it varies
at different points of a curve, must contribute to the
deep knowledge of this curve. This element is above all
very remarkable, among all the other ordinary subjects of research
in analytical geometry, that it relates directly, by its
nature, to the shape of the curve itself, without depending in any way on its
position. We see that, from the analytical point of view, it requires the
simultaneous consideration of the first two functions derived from
the ordinate.
The theory of the centers of curvature naturally leads to the important
notion of the _developed_, which are now defined as being the
geometric loci of all the centers of curvature of each curve at
its various points, although, on the contrary, in the
primitive conception of this A branch of geometry, Huyghens would have deduced the idea
of the osculating circle from that of
the
evolute , directly considered as generating by its development the primitive curve, or the _developing_. It is easy to recognize that these two ways of seeing
fit into each other. This developed obviously presents, by
whatever mode one obtains it, two general and necessary properties
relative to any curve, from which it derives: the first,
to have for tangents the normals to this one; and the second, that the
length of its arcs be equal to that of the
corresponding radii of curvature of the involute. As for the means of obtaining the equation of
the evolute of a given curve, it is clear that between the two
formulas cited above to express the coordinates of the center of
curvature, it suffices to eliminate, in each case, the x, y coordinates
of the corresponding point of the proposed curve, using the equation of
this curve: the equation in / alpha, / beta which will result from
the elimination, will be that of the developed required. One could
also undertake to solve the opposite question, that is to say to
find the involute according to the evolute. But it should be noted
that an elimination similar to the preceding one would then provide, for the
curve sought, only an equation containing, in addition to x and y, the two
derived functions dy / dx, d ^ 2y / dx ^ 2; so that after this
preparatory analysis , the complete solution of the problem would still require
the integration of this second order differential equation which,
given the extreme imperfection of the integral calculus, would most often be
impossible, if, by its own nature of such research, the curve
requested should not, as I had occasion to indicate in the
seventh lesson, be represented by the _singular_ solution, which
simple differentiation can always obtain, the general integral
designating here only the system of osculating circles, whose
knowledge is not the subject of the proposed question. It would be the
same whenever we had to determine a curve according to
any property of its radius of curvature. This order of questions
is exactly analogous to the simpler problems which constitute what
, in the origin of transcendent analysis, one called the _Inverse method
of tangents_, where one proposed to determine a curve
by a given property of its tangent at any point.
By more or less complicated geometrical considerations, analogous
to that which furnishes the evolutions, the geometers have deduced from the same
primitive curve whatever various other secondary curves,
the equations of which can be obtained by similar processes. The
most remarkable among them are the _caustics_ by reflection or
by refraction, the first idea of which is due to Tschirnaüs, although
Jacques Bernouilli alone established the true general theory. They
are, as we know, curves formed by the continual intersection
of infinitely neighboring rays of light that we would suppose reflected or
refracted by the pitch curve. Starting from the geometric law of
reflection or refraction of light, consisting in that
the angle of reflection is equal to the angle of incidence, or in that the
sine of the angle of refraction is a constant and known multiple of the
sine of the angle of incidence, it is obvious that the search for these
_caustics_ is reduced to a pure question of geometry, perfectly
similar to that of evolutes , conceived as formed by
the continual intersection of infinitely neighbors. The problem
will therefore be solved analytically by following a similar course, about
which any other indication would be superfluous here. The calculation will be
only more laborious, especially if the incident rays are not
supposed to be parallel to each other or emanate from the same point.
The evolutes, the caustics, and all the other lines deduced
from the same main curve with the aid of analogous constructions, are
formed by the continual intersections of infinitely
neighboring straight lines subject to a certain law. But we can also, by
generalizing this geometrical consideration as much as possible, conceive
of curves produced by the continual intersection of certain
infinitely close curves, subject to the same unspecified law. This
law usually consists in that all these curves are
represented by a common equation, moreover any one, from which they
derive successively by giving various values to a certain
arbitrary constant. We can then propose to find the
geometrical locus of the points of intersection of these consecutive curves, which
correspond to values infinitely close to this
arbitrary constant conceived as varying in a continuous manner. Leïbnitz was
the first to imagine research of this nature, which was then
greatly extended by Clairaut and especially by Lagrange. To treat the
simplest case, the one I have just characterized exactly, it is
obviously sufficient to differentiate the general equation proposed in relation to the
arbitrary constant that is considered, and then eliminate this
constant between this differential equation and the primitive equation;
we will thus obtain, between the two variable coordinates, an equation
independent of this constant, which will be that of the curve sought,
the shape of which will often differ greatly from that of the
generating curves . Lagrange established
an important general theorem about this geometric relation , by showing that, from the
analytical point of view , the curve thus obtained and the generating curves
necessarily have the same differential equation, the
complete integral of which represents the system generating curves, while its
_singular_ solution corresponds to the intersection curve.
So far I have considered the theory of the curvature of curves following
the spirit of the infinitesimal method proper, which
indeed adapts much more simply than any other to any research of this
kind. Lagrange's conception, relative to
transcendent analysis , presented above all, by its nature, great
special difficulties for the direct solution of such a question,
as I have already noticed in the sixth lesson. But these difficulties
so happily excited the genius of Lagrange, that they led him
to the formation of the general theory of contacts, of which the old
The theory of the osculating circle turns out to be no more than a
very simple particular case . It is important for the purpose of this work to now consider
this fine conception, which is perhaps, from a
philosophical point of view, the most profoundly interesting object that
analytical geometry can offer so far.
Let us compare any given curve y = f (x) to another
variable curve z = / varphi (x), and seek to form a precise idea of the
various degrees of intimacy which may exist between these two curves, at
a common point , according to the relations that we will suppose between the function
/ varphi and the function f. It will suffice to consider the distance
vertical of the two curves at another point closer and closer to
the first, in order to make it successively as small as possible, having
regard to the correlation of the two functions. If h denotes the increase
experienced by the abscissa on passing to this new point, this distance, which
is equal to the difference of the two corresponding ordinates, can
be developed, according to Taylor's formula, according to the
ascending powers of h, and will have for expression the series,
/ [D = / left (f '(x) - / varphi (x) / right) h +
/ left (f' '(x) - / varphi' '(x) / right) / frac {h ^ 2} {1.2} /] / [+ /left(
f''' ( x )-/ varphi''' ( x )/right)/frac Danemarkh ^3 inconnu1.2.3 } + / mbox {/ rm etc} ./]
While designing,
small, that the first term of this series is greater than the sum of
all the others, it is clear that the curve z will have a
similarity with the curve y all the more intimate, as the nature of the function
variable / varphi will allow to delete a greater number of terms
in this expansion, starting from the first. The degree of intimacy of the two
curves will therefore be exactly appreciated, from the analytical point of view,
by the greater or lesser number of successive derivative functions of
their ordinates which will have the same value at the point considered.
Hence the important general conception of the various orders of
more or less perfect _contacts_ , including the notion of the osculating circle compared to
simply tangent circles had presented only one
particular example until then . Thus, after the simple intersection, the first
degree of approximation between two curves takes place when the first
derivatives of their ordinates are equal; it is the _contact of the first
order_, or what is commonly called simple contact, because it
has long been the only one known. The _contact of the second order_ also requires
that the second derivatives of the functions f and / varphi be equal:
by again adding to it the equality of their third derivatives, we
constitute a _contact of the third order_, and so on ad infinitum.
Beyond the first order, contacts often bear the name
of first order, second order osculations, etc.
The contacts of the first and second order can be characterized
geometrically by a very simple observation, in that it
obviously follows that the two curves compared have at the common point, in one
case, the same tangent, and, in the other , the same circle of curvature,
since the tangent to each curve depends on the first derivative of its
ordinate, and the circle of curvature, on the first two
successive derivatives . But this consideration would no longer be appropriate beyond the
second order to determine the geometric idea of contact. Lagrange
confined himself, in this respect, to assigning the general character
results immediately from the analysis indicated above, and which consists
in that when the curve z is determined so as to have
a contact of the order n with the curve y, product analytically by the equality
of all the derived functions up to that of order n, no
other curve z, of the same nature as the preceding one, but which would
satisfy only a smaller number of analytical conditions, and which,
consequently, would have with the curve y that 'a less intimate contact,
could not pass between the two curves, since the interval of
these received the smallest value of which it was susceptible according to
such a relation of the two equations.
When we have particularized the nature of the curve z thus compared to
any given curve y, the order of the most intimate contact that it
can have with it obviously depends on the greater or lesser number
of arbitrary constants contained in its most general equation, a
contact of the order n requiring n + 1 analytical conditions, which can only
be fulfilled with such a number of
available constants . Therefore, a straight line, whose most
general equation contains only two arbitrary constants, can have
with any curve only a contact of the first order: from which follows
the ordinary theory of tangents. The equation of the enclosing circle, in
general, three arbitrary constants, the circle can have with
any curve a contact of the second order, and from this results, as a
particular case, the old theory of the osculating circle. In
considering a parable, as there are four arbitrary constants
in its fullest and simplest equation, it is
likely, compared to other curved, more privacy
deep, who can go to the contact of the third order : similarly,
an ellipse would have a contact of the fourth order, etc.
The preceding consideration is apt to suggest a
geometric interpretation of this general theory of contacts, which seems to me
intended to supplement the work of Lagrange, by assigning, to
directly define the various orders of contacts, a concrete character
simpler and clearer than that indicated by Lagrange. Indeed, this
more or less large number of arbitrary constants contained in an
equation has for geometric significance, as we established at the
beginning of this lesson, the number of points necessary for the entire
determination of the corresponding curve, which is found thus
mark the degree of intimacy of which this curve is susceptible
relative to any other. However, on the other hand, the analytical law which
expresses this contact by the equality of such a number of derivatives
of the two ordinates, obviously indicates that the two curves
then have as many infinitely neighboring points in common; since, from
the nature of the differentials, it is clear that the differential of
order n depends on the comparison of n + 1 consecutive ordinates. We
can therefore directly get a clear idea of the various orders of
contacts, by saying that they consist in the community of a
greater or lesser number of infinitely neighboring points between the two curves.
In more rigorous terms, we would define, for example, the
third-order osculating ellipse , looking at it as the limit towards
which the ellipses passing through five points of the curve would tend.
proposed, as four of these supposedly mobile points
approach indefinitely the fifth supposedly fixed.
This general theory of contacts is evidently proper, by its
nature, to provide an increasingly deep knowledge of the
curvature of any curve, by successively comparing
various known curves with it, susceptible of more and more
intimate contact; which would make it possible to make the
measurement of the curvature as exact as one would like , by suitably changing the
comparison term . Thus, it is clear, from the
preceding considerations , that the assimilation of any arc of infinitely small curve to
an arc of a parabola, would make its curvature known with more than
precision only by the use of the osculating circle; and the comparison with
the ellipse would provide even more accuracy, etc .; so that by
destining each primitive type to deepen the study of the following type, we
could improve the theory of curves ad infinitum. But the
necessity of having a clear and familiar knowledge of the curve thus
adopted as the unit of curvature, determines the surveyors to renounce
this high speculative perfection, to be satisfied, in reality, to
compare all the curves with the circle only, by virtue of of
the curvature of uniformity characteristic property of the circle. No
other curve, in fact, can be regarded, in this respect, as
simple enough and well-known enough to be usefully employed,
although we no longer ignore that the circle is not
the most suitable unit of courbure in the abstract. Lagrange therefore confined himself
definitively to deducing from his general conception the theory of the
osculating circle , thus presented from a purely analytical point of view. It
is even remarkable that from this consideration alone he was able to conclude
with ease the two fundamental properties above indicated for
the evolutions, which simple analysis at first seemed so little adapted
to establish.
I thought I should consider the theory of the contacts of curves in its
greater speculative extension, in order to properly grasp
its true character. Although we must ultimately reduce it to the
sole effective determination of the osculating circle, there is undoubtedly,
from a philosophical point of view, a profound difference between conceiving
this last consideration, so to speak, as the last term
of the efforts of the mind. human in the study of curves, as was
done before Lagrange, and to see there, on the contrary, only a simple
particular case of a very extensive general theory, to the examination of which one
must usually limit, knowing nevertheless that other
comparisons could further improve the geometrical doctrine.
After having considered the main questions of general geometry
relating to the properties of curves, it remains for me to point out those which
relate to rectifications and quadratures, in which
properly consists, according to the explanation given in the tenth lesson,
the final goal of the geometric science. But having had occasion
previously (_see_ the 6th lesson) to establish the general formulas
which express, with the aid of certain integrals, the length and the area
of any plane curve of which the rectilinear equation is given, and
before besides, forbid me here any application to any
particular curve , this important part of the subject is sufficiently
processed. I will confine myself only to indicating the formulas suitable for
determining the area and volume of bodies produced by the revolution of
plane curves around their axes.
Let us suppose, as one can obviously always do it, that the axis of
rotation is taken for axis of the abscissas; and, following the spirit of the
infinitesimal method properly so called, the only one well suited up to
now for research of this nature, let us conceive that the abscissa
increases by an infinitely small quantity: this increase will determine
in the arc and in the area of the curve of the
analogous differential increases which, by the revolution around the axis, will generate the
_elements_ of the desired surface and volume. It is easy to see that,
by neglecting only an infinitely small of the second order at
most, we will be able to regard these elements as equal to the area and to the
volume of the truncated cone or of the corresponding cylinder, having for height
the differential of the abscissa, and for radius of its base the ordinate of the
point considered. According to this, by calling S and V the
requested area and volume, the simplest propositions of
elementary geometry will immediately provide the
general differential equations / [dS = 2 / pi ydx, /; dV = / pi y ^ 2dx. /] Thus, when the relation
between y and x is given in each particular case, the values of S
and of V will be expressed by the two integrals / [S = 2 / pi / int
yds, /; V = / pi / int y ^ 2dx; /] taken between the appropriate limits. Such
are the invariable formulas according to which, since Leibnitz,
geometers have solved a great number of questions of this kind, when the
progress of integral calculus has permitted it.
One could also include among the number of researches in
general two-dimensional geometry , the important determination of the centers of
gravity of arcs or of the areas belonging to any curves,
although this consideration has its origin in
rational mechanics . Because, by defining the center of gravity as being the
_centre of the mean distances_, that is to say a point whose distance
to a plane or to any axis is the arithmetic mean between the
distances of all the points of the body to this plane or to this axis, it is
clear that this question becomes purely geometric, and can be
dealt with without any recourse to mechanics. But, despite such a
consideration, the importance of which we shall later recognize in order to
generalize sufficiently and easily the notion of the center of
gravity, it is certain, on the other hand, that the
essential destination of this research must continue to be. classify more
appropriately among questions of mechanics; although by its nature
proper, and also by the analytical character of the
corresponding method , it really belongs to geometry, which prompted me
to indicate it here by anticipation.
These are the main fundamental questions of which the
present system of our general two-dimensional geometry is composed . We see
that, from the analytical point of view, they can be clearly
distinguished into three classes: the first, comprising
geometrical researches which depend only on ordinary analysis; the second,
those whose solution requires the use of differential calculus; the
third, finally, those which can only be solved using the
integral calculus.
It remains for us now to consider under the same aspect, in the
following lesson , the whole of general geometry in three dimensions.
FOURTEENTH LESSON.
SUMMARY. Three-dimensional _general_ geometry.
The study of surfaces is made up of a series of general questions
exactly analogous to those indicated in the previous lesson in
relation to lines. It is useless to consider here separately
those which depend only on ordinary analysis, for they are
resolved by essentially similar methods; either it
is a question of knowing the number of points necessary for the entire
determination of a surface, or that one deals with the search for
centers, or we ask for the precise conditions of the similarity
between two surfaces of the same kind, etc. There is no other
analytical difference than to consider equations with three variables instead
of equations with two variables. I therefore pass immediately to the questions
which require the use of transcendent analysis, insisting only
on the new considerations which they present in relation to
surfaces.
The first general theory is that of tangent planes. By using
the infinitesimal method itself, one can easily find
the equation of the plane which touches any surface at a given point,
infinitely small extent all around the point of contact. It suffices, in
fact, to consider that, in order to fulfill such a condition,
the infinitely small increase received by the vertical ordinate as a
result of the infinitely small increases of the two
horizontal coordinates , must be the same for the plane as for the surface. , and
that independently of any definite relation between these two last
increases, without which the coincidence would not take place in any direction.
According to this idea, the analysis immediately gives the general equation:
/ [zz '= / frac {dz'} {dx '} (x-x') + / frac {dz '} {dy'} (y -y ') /] for that of
the tangent plane, x', y ', z',
The determination of this plane, in each particular case, is
thus reduced to a simple differentiation of the equation of the
proposed surface .
One can also obtain this general equation of the tangent plane, by
making depend its research only on the theory of tangents to
plane curves. For this, we must consider this plane, as
is usually done in descriptive geometry, as determined by the
tangents to any two plane sections of the surface passing at
the given point. By choosing the planes of these sections parallel to two
of the coordinated planes, we immediately arrive at the preceding equation.
This way of conceiving the tangent plane gives rise to
easily an important theorem of general geometry, which Monge was
the first to demonstrate, and which consists in the fact that the tangents to all
the curves which can be drawn at the same point on
any surface are always included in the same plane.
Finally, it is still possible to arrive at the general equation of the
tangent plane by considering it as perpendicular to the
corresponding normal , and defining this by its
direct geometric property of being the _maximum_ or _minimum_ path to go from a point
outside to the surface. The ordinary method of _maxima_ and _minima_
suffices to form, according to this notion, the two equations of the
normal, by applying this method to the expression of the distance
between two points, one located on the surface, the other outside, of which
the first conceived as variable, is then supposed fixed when the
analytical conditions have been expressed, while that the second,
originally constant, is then considered as mobile, and describes the
line sought. The equations of the normal once obtained, one
easily deduces from it that of the tangent plane. This ingenious way of
establishing it is also due to Monge.
The fundamental question which we have just examined becomes, as in
the case of curves, the basis of a large number of studies relating to
the determination of the tangent plane, when one replaces the point of contact
given by other equivalent conditions. The tangent plane
obviously can not be determined by a single given external point, as
is the tangent: it must be subjugated to contain a given straight line; except
that the analogy is perfect, and the two questions are resolved
in the same way. It is the same if the tangent plane must be
parallel to a given plane, which fixes the value of the two constants which
assign its direction, and consequently determines the coordinates of the point
of contact, of which these constants are, for each designated surface ,
known functions. Finally we can also find as in the curves,
the analytical relation which generally expresses the simple phenomenon of the
contact between a plane and a surface, without specifying the place of this
contact; from which similarly results the solution of several questions
relating to tangent planes, among others that which consists in determining
a plane which touches at the same time any three given surfaces, a
research analogous to that of the common tangent of two curves.
The general theory of the more or less intimate contacts which can
exist between any two surfaces as a result of the more or
less numerous relations of their equations, is formed according to a method
exactly similar to that indicated in the preceding lesson.
relative to the curves, by expressing, using the Taylor series
for the functions of two variables, the vertical distance of the two
surfaces at a second point close to their point of intersection, and whose
horizontal coordinates would have received two increases h and k
completely independent of each other. The consideration of this
distance, developed according to the increasing powers of h and k, and in
the expression of which we will successively remove the terms of the
first degree in h and k, then those of the second, etc., will determine the
analytical conditions of the contacts of different orders that
the two surfaces can have depending on the greater or lesser number of
arbitrary constants contained in the general equation of that which one
regards as variable. But, in spite of the conformity of method, this
theory will present with that of curves a fundamental difference
relatively to the number of these conditions, owing to the necessity in which
we find ourselves in this case to consider two independent increases
instead of a single one. . It follows, in fact, that, in order for each contact to
take place in all possible directions around the common point, we must
separately cancel all the different terms of the same
corresponding degree , and, the number of which will increase all the more as this degree
or order of contact will be higher. So after the condition of
the equality of the two vertical ordinates z necessary for the simple
intersection, one will find that the contact of the first order requires,
moreover, two distinct relations, consisting in the respective equality
of the two partial derivative functions of the first order specific to each
vertical ordinate . While passing to the contact of the second order, it will
be necessary to add three more new conditions, because of the three
distinct terms of the second degree in h and k in the expression of the distance, and
whose complete elimination will require the respective equality of the three
functions second order partial derivatives relative to the z of each
surface. We will find in the same way as the contact of the third
order gives rise in addition to four other relations, and so on,
the number of partial derivatives of each order remaining constantly
equal to the number of terms in h and k of the corresponding degree. It is easy
to conclude, in general, that the total number of distinct conditions
necessary for the contact of the order n, has for value (n + 1) (n + 2) / 2, while
in the curves, it was simply equal to n + 1.
As a consequence of this only essential difference, the theory of surfaces
is far from offering in this respect the same facility and comprising the same
perfection as that of curves. When we limit ourselves to the contact of the
first order, there is complete parity, since this contact requires only
three conditions, which can always be satisfied using the
three arbitrary constants contained in the general equation of a plane;
from this results, as a particular case, the theory of tangent planes,
exactly analogous to that of tangents to curves, and presenting the
same utility for studying the shape of any surface. But
this is no longer the case when the second order contact is considered, in order
to measure the curvature of the surfaces. It would be natural then to compare
all the surfaces with the sphere, the only one which presents a
uniform curvature , as one compares all the curves with the circle. However, the
second order contact between two surfaces requiring six conditions, while
the most general equation of a sphere contains only four
arbitrary constants, it is not possible to find, at each
point of any surface, a sphere which is completely
osculating in all directions, instead that we have seen an
infinitely small arc of curve can always be assimilated to a certain arc of a
circle. From this impossibility of measuring the curvature of a surface
at each point using a single sphere, geometers determined
the coordinates of the center and the radius of a sphere which, instead of being
osculating in any meaning indiscriminately, would be so only in a
certain particular direction, corresponding to a given relation between
the two increases h and k. It is then sufficient, in fact, to establish
this _relative_ second order contact, to add, to the three
ordinary conditions of the first order contact, the unique condition which results
from the total suppression of the second degree terms in h and k considered
collectively , without it being necessary to cancel each
separately; the number of relations is thus found only equal to
that of the available constants contained in the general equation of
the sphere, which is thus determined. This process is properly reduced to
studying the curvature of a surface at each point by that of the
various curves that would be traced on this surface by a series of planes.
led by the corresponding normal.
According to the general formula which expresses the radius of curvature of each
of these normal sections as a function of its direction, Euler, to whom
this whole theory is essentially due, discovered several
important theorems relating to any surface. He at first easily
established that, among all the normal sections of a surface at the same
point, one could distinguish two principal ones, of which the curvature,
compared to that of all the others, was a _minimum_ for the
first, and a _maximum_ for the second, and the planes of which present
the remarkable circumstance of being constantly perpendicular between
them. He then showed that, whatever the surface
proposed, and without it being even necessary to define it, the curvature
of these two main sections was sufficient alone to
completely determine that of any other normal section, at l Using
an invariable and very simple formula, according to the inclination of the plane
of this section on that of the section of greater or
lesser curvature. By considering this formula as the polar equation
of a certain plane curve, he deduced from it an ingenious construction,
eminently remarkable for its generality and for its simplicity. It
consists in that, if we construct an ellipse such that the
distances from one of its foci at the two ends of the major axis are
equal to the two radii of curvature _maximum_ and _minimum_, the radius of
curvature of any other normal section will be equal to that of the
vector radii of the ellipse which will form with the axis an angle double of
the inclination of the plane of this section on that of one of the
main sections . This ellipse changes into a hyperbola constructed in the
same way, when the two main sections do not turn their
concavity in the same direction: finally it becomes a parabola, when the
surface is of the kind that can be generated by a
straight line , or that it presents an _inflexion_ at the point considered.
From this beautiful fundamental property, we have later concluded a large
number of more or less interesting secondary theorems, which it is
not the place to indicate here. I must only point out the
essential theorem by which Meunier completed the work of Euler, by
relating the curvature of all the arbitrary curves which can
be traced on a surface at the same point, with that of the
normal sections , the only ones that Euler would have considered. This theorem consists in
that the center of curvature of any oblique section can be considered
as the projection on the plane of this section, of the center of curvature
corresponding to the normal section which would pass through the same tangent:
from which Meunier deduced a very simple construction, according to which,
by the use of a circle analogous to the ellipse of Euler, one determines the
curvature of the oblique sections, knowing that of the normal sections;
so that, by the combination of the two theorems, the only curvature
of the two _principales_ normal sections suffices to obtain that of
all the other curves which can be drawn on a surface in
any way at each point considered.
The previous theory allows to study completely, point by point, the
curvature of any surface. In order to link together more easily
the considerations relating to the various points of the same surface,
the geometers sought to determine what they call the _lines
of curvature_ of a surface, that is to say, those which enjoy the
property that the consecutive normals to the surface can be
regarded as included in the same plan. At each point of
any surface, there are two such lines, which are found to be
constantly perpendicular to each other, and whose directions
coincide at their origin with those of the two
_principales_ normal sections considered above, which may dispense with 'consider
the latter separately. The determination of these lines of curvature is
carried out very simply on the most usual surfaces, such as
as cylindrical, conical, and revolving surfaces. This
new fundamental consideration has moreover become the
starting point of several other less important general researches, such as
that of _surfaces of curvature_, which are the geometric loci of the
centers of curvature of the various _principales_ sections; that of
the developable surfaces formed by the normals to the surface conducted at the
various points of each line of curvature, etc.
To complete the examination of the theory of curvature, it remains for me to
indicate briefly what relates to _curves with double
curvature_, that is to say, to those which cannot be contained in a
plane.
As for the determination of their tangents, this obviously offers
no difficulty. If the curve is given analytically by the
equations of its projections on two of the coordinate planes, the
equations of its tangent will simply be those of the tangents to these
two projections, which brings the question into the case of
plane curves. If, from a more general point of view, the
analytical definition of the curve consists, as the twelfth lesson indicates,
in the system of equations of any two surfaces of which it
is the intersection, we will regard the tangent as being
l 'intersection of the tangent planes to these two surfaces, and the problem
will be reduced to that of the tangent plane, solved above.
The curvature of curves of this nature gives rise to the establishment
of a very important new notion. In fact, in a plane curve,
the curvature is found to be sufficiently appreciated by measuring
the greater or lesser inflection of the consecutive elements on each
other, which is estimated indirectly by the radius of the osculating circle.
But it is not at all thus in a curve which is not plane.
The consecutive elements not being then any more contained in the same plane,
one can have an exact idea of the curvature only by considering
distinctly the angles which they form between them and also the
mutual inclinations of the planes which include them. It is therefore necessary,
above all, to begin by fixing what one must understand at each instant
by _the plane_ of the curve, that is to say, that determined by three
infinitely neighboring points, and which we call, for this reason, the
_osculator_ plane , which changes continuously from one point to another. The
position of this plane once obtained, the measurement of the curvature
usually with the osculating circle, no longer obviously
no new difficulty. As for the second curvature, it is
estimated by the greater or lesser angle formed between them two
consecutive osculating planes , and which is generally easy to find.
analytical expression. To establish more analogy between the theory
of this curvature and that of the first, one could also
regard it as measured indirectly according to the radius of the
_osculatory_ sphere which would pass through four points infinitely close to the
proposed curve, and of which l The equation would form in the same way as
that of the osculating plane. It is usually appreciated by the
maximum curvature that presents, at the point considered, the developable surface which
is the geometric locus of all the tangents to the proposed curve.
We must now move on to the indication of questions of
general three-dimensional geometry which depend on the integral calculus; they
include the squaring of curved surfaces, and the cubature of the
corresponding volumes.
Relatively to the quadrature of curved surfaces, to establish
the general differential equation , it is necessary to conceive of the surface divided into
elements planes infinitely small in all directions, by four planes
perpendicular two by two to the axes of the coordinates x and y. Each of
these elements, located in the corresponding tangent plane, would obviously have
for horizontal projection, the rectangle formed by the differentials
of the two horizontal coordinates, and whose area would be dxdy. This
area will give that of the element itself, according to a theorem
elementary very simple, by dividing it by the cosine of the angle
made by the tangent plane with the x, y plane. We will thus find that
the expression of this element is generally: / [d ^ 2S =
dxdy / sqrt {/ frac {dz ^ 2} {dx ^ 2} + / frac {dz ^ 2} {dy ^ 2} +1 } /] It is therefore by the
double integration of this differential formula with two variables
that we will know, in each particular case, the area of the
proposed surface , as much as the current imperfection of the
integral calculus will be able to allow . The limits of each successive integral will be
determined by the nature of the surfaces whose intersection with
the one considered should circumscribe the extent to be measured, so that,
in applying this general method,
special care should be taken in how to fix arbitrary constants or
arbitrary functions introduced by integration.
Relative to the cubature of volumes terminated by
curved surfaces , the system of planes with the help of which we have just differentiated
the area, can also be used immediately to decompose the volume into
polyhedron elements. It is clear, in fact, that the infinitely small space
of the second order included between these four planes, must be considered,
according to the spirit of the infinitesimal method, as equal to the
rectangular parallelepiped having for height the vertical ordinate z of the
point which one considers and for base the rectangle dxdy, since their
difference is obviously an infinitely small of the third order,
less than dzdydz. According to this, one of the simplest theorems of
elementary geometry will provide directly, for the
differential expression of the sought volume, the general equation / [d ^ 2V = zdxdy; /]
from which we will deduce, by a double integration, in each
particular case , the effective value of this volume, having the same regard
as previously to the determination of the limits of each integral, in
accordance with the nature of the surfaces which must
laterally circumscribe the proposed volume.
Without going into any detail here relating to the final solution of
one or the other of these two fundamental questions, it may be
useful to notice, according to the preceding differential equations,
a general and singular analogy which necessarily exists between
them, and which would make it possible to transform any research relating to the
quadrature into a corresponding search for cubature. We
see, in fact, that the two differential equations differ only
by the change of z in / sqrt {dz ^ 2 / dx ^ 2 + dz ^ 2 / dy ^ 2 + 1} while passing from the
second to the first. Thus the area of any curved surface can
be regarded as numerically equal to the volume of a body terminated by
a surface whose vertical ordinate would have at each instant for
value the secant of the angle made with the horizontal plane by the
tangent plane corresponding to the primitive surface, the limits being
moreover respectively assumed to be the same.
To complete the philosophical examination of general three-
dimensional geometry , it remains for me to briefly consider the beautiful
fundamental conception established by Monge with regard to the
analytical classification of surfaces into natural families, which must be regarded
as the most important improvement that has received
geometrical science from Descartes and Leïbnitz.
When we propose to study, from a general point of view, the
special properties of various surfaces, the first difficulty which
presents itself consists in the absence of a good classification,
determined by the most essential geometric characteristics, and
moreover sufficiently simple. From the founding of
analytical geometry , geometers were unwittingly led to classify
surfaces, like curves, by the form and degree of their
equations, the only consideration which offered itself to the mind to
serve basic to a distinction the importance of which had not at first been
felt. But it is easy to see that this principle of
classification, suitably applicable to first and
second degree equations , does not meet any of the main conditions to which
satisfy such work. Indeed, we know that Newton, in discussing
the general equation of the third degree in two variables, to limit himself
to the simple enumeration of the various plane curves which it can
represent, recognized that, although they were all necessarily
undefined in every sense, one could distinguish 74
particular species , as different from each other as
the three curves of the second degree are from one another. Although no one has analyzed
from the same point of view the general equation of the fourth degree in two
variables, there is no doubt that it must have given rise to a
still more considerable number of distinct curves; and this number
should obviously increase with prodigious rapidity according to the
degree of the equation. If we now pass to the equations with three
variables, which, given their greater complexity,
necessarily present much more variety, it is indisputable that the number
of truly distinct surfaces which they can express must be
still more multiplied, and increased. much faster depending on the
degree. This multiplicity becomes such that we have always confined ourselves to
analyzing the equations of the first two degrees in this way, no geometer
having attempted for the surfaces of the third degree what
Newton did for the corresponding curves. So it follows from this
an evident consideration that, even if the imperfection of algebra did
not preclude the indefinite use of a similar process, the
general classification of surfaces by the degree and form of their
equations would be entirely impracticable. But this reason is not the
only one which should make reject such a classification; it is not
even the most important. In fact, this way of arranging
surfaces, apart from the impossibility of following it, is directly
contrary to the main purpose of any good classification
whatever, consisting of bringing the
objects which offer the most important relationships as close as possible to each other. , and to move away
those whose analogies are of little value. The identity of the degree of their
equations is, for surfaces, a character of
very mediocre geometric value , which does not even indicate exactly the number of points
necessary for the entire determination of each. The
most important common property to consider between surfaces obviously consists
in their mode of generation; all those which are generated in the
same manner must necessarily offer a great
geometrical analogy , while they can only have very weak
resemblances if they are generated according to essentially
different modes . So, for example, all cylindrical surfaces, which
whatever the shape of their base, constitute the same natural family,
the various species of which present a large number of
common properties of prime importance: it is the same for all
conical surfaces, and also for all surfaces of revolution, etc. .
Now, this natural order is completely destroyed by the
classification based on the degree of the equations. For surfaces
subject to the same mode of generation, cylindrical surfaces, for
example, can provide equations of all conceivable degrees,
at the rate of the only secondary difference of their bases; while,
on the other hand, equations of any given degree express
often surfaces of opposite geometrical nature, some
cylindrical, others conical, or of revolution, etc. Such an
analytical classification is therefore radically flawed, as separating
what must be brought together, and bringing together what must be distinguished.
However, the general geometry being entirely based on the use
of considerations and analytical methods, it is essential that
the classification can also take on an analytical character.
Such was therefore the precise state of the fundamental difficulty, so
happily overcome by Monge: the natural families between
surfaces being clearly established under the geometric point of view
according to the mode of generation, it was necessary to discover a kind of
analytical relationship intended to constantly present an
abstract interpretation of this concreteness. This capital discovery was
absolutely essential to complete the constitution of the
general theory of surfaces.
The consideration, which Monge employed to achieve this, consists in
this general observation, as simple as it is direct: surfaces
subject to the same mode of generation are necessarily
characterized by a certain common property of their tangent plane at
any point; so that by expressing this
property analytically according to the general equation of the plane tangent to a surface
arbitrary, we will form a differential equation representing at the same
time all the surfaces of this family.
Thus, for example, any cylindrical surface presents this
exclusive character : that the tangent plane at any point of the surface is
constantly parallel to the fixed line which indicates the direction of the
generatrices. From this, it is easy to see that the equations of
this line being supposed to be / [x = az, /; y = bz, /] the general equation
of the tangent plane established above will give, for the equation
differential common to all cylindrical surfaces,
/ [a / frac {dz} {dx} + b / frac {dz} {dy} = 1. /]
Similarly, relatively to conical surfaces, they are all
characterized from this point of view by the necessary property that their
tangent plane at any point constantly passes through the apex of the
cone. If therefore / alpha, / beta, / gamma, denote the coordinates of this
vertex, we will immediately find / [(x- / alpha) / frac {dz} {dx} +
(y- / beta) / frac {dz} { dy} = z- / gamma, /] for the differential equation
representing the entire family of conical surfaces.
In surfaces of revolution, the tangent plane at any point
is always perpendicular to the _meridian_ plane, that is to say to that
which passes through this point and the axis of the surface. In order to
analytically express this property in a simpler way, suppose that
the axis of revolution is taken for that of z: the
differential equation common to all this family of surfaces, will be
/ [y / frac {dz} {dx} -x / frac {dz} {dy} = 0. / ]
It would be superfluous to cite here a greater number of examples in order to
establish clearly, in general, that, whatever the mode of
generation, all the surfaces of the same natural family are
capable of being represented analytically by the same one. equation
_at partial differences_ containing arbitrary constants,
according to a common property of their tangent plane.
In order to complete this fundamental and necessary correspondence between
the geometrical point of view and the analytical point of view, Monge has
considered in addition the finite equations which are the integrals of these
differential equations, and which can moreover almost always
easily be obtained also by direct research. Each of these
finite equations must, as we know from the general theory of
integration, contain an arbitrary function, if the
differential equation is only of the first order; This does not prevent
such equations, although much more general than those with which we are
ordinarily concerned, from presenting a clearly determined meaning, either
under the geometric relation, or under the simple analytical relation.
This arbitrary function corresponds to what is indeterminate in
the generation of the proposed surfaces, at the base, for example, if the
surfaces are cylindrical or conical, at the meridian curve, if they
are of revolution, etc. [27]. In some cases even, the finite equation
of a family of surfaces contains at the same time two arbitrary functions,
assigned to distinct combinations of the variable coordinates;
this is what takes place when the corresponding differential equation must be
of the second order; under the geometric point of view, this
greater indeterminacy indicates a more general family, and nevertheless
characterized. Such is, for example, the family of
developable surfaces , which includes, as subdivisions, all surfaces
cylindrical, all conical surfaces, and an infinity of other
similar families, and which can however be clearly defined, in
its greatest generality, as being the_envelope_ of the space
traversed by a plane which moves while remaining always tangent with any two
fixed surfaces, or as the geometrical locus of all the
tangents to the same unspecified curve with double curvature. This
natural group of surfaces has, for invariable differential equation, this
very simple equation, discovered by Euler, between the three
partial derivatives of the second order, / [/ left (/ frac {d ^ 2z} {dxdy} / right) ^ 2 =
/ frac {d ^ 2z} {dx ^ 2} / frac {d ^ 2z} {dy ^ 2} /].
[Note 27: We find, for example, either from
direct considerations of analytical geometry, either as a
result of the integration methods, that the
cylindrical surfaces and the conical surfaces have as
finite equations / x-az = / varphi (y-bz), /; / frac {x- / alpha} {z- / gamma} =
/ varphi / left (/ frac {y- / beta} {z- / gamma} / right) / / varphi,
denoting an entirely arbitrary function.]
L ' finite equation therefore necessarily contains two
distinct arbitrary functions , which correspond geometrically to the two
indeterminate surfaces on which the generating plane must slide, or to
any two equations of the directing curve.
Although it is useful to consider the finite equations of families
natural surfaces, it is nonetheless conceivable that the indeterminacy of the
arbitrary functions which they inevitably contain must
make them hardly suitable for sustained analytical work, for which it
is much preferable to use differential equations, in which
only simple arbitrary constants, despite their
indirect nature . It is by this that the general and regular study of the properties
of the various surfaces has really become possible, the
common point of view having thus been able to be grasped and separated by analysis. It is
understandable that such a conception made it possible to discover results of a degree
of generality and interest infinitely greater than those which could be
get before. To cite only one very simple example, which
is far from being the most remarkable, it is by a similar
method of analytical geometry that we have been able to recognize this singular
property of any _homogeneous_ equation with three variables, to
necessarily represent a conical surface whose vertex is located at the origin
of the coordinates; similarly, among the more difficult searches, it has
been possible to determine, using the calculation of variations, the
shortest path from one point to another on
any developable surface , without it being necessary to particularize, etc.
I thought it my duty here to give some development to the exhibition
philosophy of this beautiful conception of Monge, which constitutes, without a
doubt, his first title to glory, and whose great importance does
not seem to me to have yet been worthily felt, except by Lagrange,
so just appreciator of all his followers. I even regret to be
reduced, by the natural limits of this work, to such an
imperfect indication , in which I could only point out the happy
necessary reaction of this new geometry on the improvement of
the analysis, as for the general theory of differential equations with
several variables.
By meditating on this philosophical classification of surfaces,
essentially analogous to the natural methods that physiologists
have attempted to establish in zoology and botany, one is led to
wonder whether the curves themselves do not involve a
similar operation . Considering the infinitely less variety which exists between them,
such a work is at the same time less important and more difficult, the
characters which could be used as a basis then not being very
much so decided. It was therefore natural that the human mind should
first concern itself with classifying surfaces. But one must doubtless hope that this
order of considerations will extend later to the curves. We can
even already see some truly natural families between them,
such as that of any parabolas, and that of hyperbolas.
any, etc. Nevertheless, no
general conception directly suitable for determining such a classification has yet been produced .
Having thus explained as clearly as I could, in this
lesson and in the whole of the four preceding ones, the true
philosophical character of the most general and the simplest section of
concrete mathematics, I must now undertake the same work
relative to the immense and more complicated science of
rational mechanics . This will be the subject of the following four lessons.
FIFTEENTH LESSON.
SUMMARY. Philosophical considerations on the fundamental principles
of rational mechanics.
Mechanical phenomena are, by their nature, as we have already
noticed, at the same time more particular, more complicated and more concrete
than geometric phenomena. Also, in accordance with the
encyclopedic order established in this work, let us place
rational mechanics after geometry in this philosophical exposition of
concrete mathematics, as being necessarily of a more
difficult study , and consequently less perfect. Geometric questions
are always completely independent of any
mechanical consideration , while mechanical questions are
constantly complicated by geometric considerations, the shape of bodies in front of
inevitably influence the phenomena of movement or
balance. This complication is often such that the simplest
change in the shape of a body is enough alone to
extremely increase the difficulties of the mechanical problem of which it is the
subject, as one can get an idea by considering, for example,
the important determination of the mutual gravitation of two bodies as a
result of that of all their molecules, a question which is not yet
fully resolved except by supposing that these bodies have a spherical shape, and
where, therefore, the main obstacle obviously comes the
geometric circumstances.
Since we have recognized in the previous lessons that the character
philosophy of geometrical science was still altered to a certain
degree by a very perceptible remnant of the influence of the metaphysical mind,
one must naturally expect, in view of this greater
necessary complication of rational mechanics, to find it. much more
deeply affected. This is, in fact, all too easy to
see. The character of natural science, even more evidently
inherent in mechanics than in geometry, is today completely
disguised in almost all minds, by the use of
ontological considerations . We notice, in all the fundamental notions of
this science, a deep and continual confusion between the point of
abstract view and the concrete point of view, which prevents one from
clearly distinguishing what is really physical from what is purely logical,
and from separating with exactitude the artificial conceptions only
intended to facilitate the establishment of the general laws of equilibrium
or movement, natural facts provided by effective observation
of the outside world, which constitute the real basis of science. We
can even recognize that the immense improvement of
rational mechanics over the past century, either in terms of the extension of its
theories, or in terms of their coordination, has in a way caused
the philosophical conception of science to retrograde in this respect. ,
which is commonly exhibited today in a much less
clear manner than Newton had presented it. This development having been, in
fact, essentially obtained by the more and more exclusive use of
mathematical analysis, the preponderant importance of this admirable
instrument has gradually made us contract the habit of seeing in
rational mechanics only simple analytical questions; and, by an
abusive, although very natural, extension of such a way of
proceeding, an attempt has been made to establish, _a priori_, from
purely analytical considerations , down to the fundamental principles of science,
which Newton s '
only observation. It is thus, for example, that Daniel Bernouilli,
d'Alembert, and, nowadays, Laplace, tried to prove the
elementary rule of the composition of the forces by
purely analytical demonstrations , of which Lagrange alone saw well the
radical and necessary insufficiency . Such is, even now, the spirit which
more or less dominates in all geometers. It is nevertheless evident in
general thesis , as we have several times observed, that
mathematical analysis , whatever its extreme importance, of which I have tried to
give a fair idea, cannot be, by its nature, only '' a powerful
means of deduction, which, when applicable, allows
to perfect a science to the most eminent degree, after the
foundations have been laid, but which can never suffice to establish
these foundations themselves. If it were possible to constitute the
science of mechanics entirely from simple analytical conceptions, one
could not imagine how such a science would ever
really become applicable to the effective study of nature. What establishes the
reality of rational mechanics is precisely, on the contrary,
to be founded on a few general facts, immediately furnished by
observation, and which every truly positive philosopher must consider,
it seems to me, as n ' being susceptible to no explanation
any. It is therefore certain that the
analytical mind has been abused in mechanics , much more than in geometry. The special object of
this lesson is to indicate how, in the present state of science, one
can clearly establish its true philosophical character, and
definitively free it from all metaphysical influence, by
constantly distinguishing the abstract point of view from the point of view. concrete point of view, and by
making an exact separation between the merely
experimental part of science, and the purely rational part. According to
the purpose of this work, such work must necessarily precede
general considerations on the effective composition of this science,
which will be successively exposed in the following three lessons.
Let us begin by indicating with precision the general object of science.
We are accustomed to notice first, and with great reason, that
mechanics does not consider not only the first causes of
movements, which are outside all positive philosophy, but even
the circumstances of their production. which, although
really constituting an interesting subject of positive research in the
various parts of _physics_, are by no means the domain of
mechanics, which confines itself to considering the movement in itself, without
inquiring how it was determined. Thus the _forces_ do not
are something else, in mechanics, than the movements produced or tending to
be produced; and two forces which impart to the same body the same
speed in the same direction are regarded as identical, however
diverse their origin may be, whether the movement comes
from the muscular contractions of an animal, or from gravity towards an
attractive center , or the shock of any body, or the expansion
of an elastic fluid, etc. But, although this view has
fortunately become quite familiar today, it still remains
for geometers to make, if not in the conception itself, at least in the
usual language, an essential reform in order to completely rule out
the old metaphysical notion of _forces_, and indicate more clearly
than is still the true point of view of mechanics [28].
[Note 28: It is important to note also that the very name
of the science is extremely vicious, in that it
only recalls one of its most secondary applications,
which usually becomes a source of confusion, which
requires the frequent addition of l 'rational adjective,
the repetition of which, although essential , is tedious. The
German philosophers, to avoid this inconvenience,
created the much more philosophical denomination of
_phoronomie_, employed in the treaty of Hermann, and of which
general adoption would be very desirable.]
This being said, we can characterize in a very precise manner the
general problem of rational mechanics. It consists in determining the effect
which will produce on a given body various unspecified forces
acting simultaneously, when one knows the simple movement which
would result from the isolated action of each of them; or, taking the
question in the opposite direction, to determine the simple motions the
combination of which would give rise to a known compound motion. This statement
shows exactly what are necessarily the data and the
unknowns of any mechanical question. We see that the study of the action
of a single force is never, strictly speaking, in the domain of
rational mechanics, where it is always supposed to be known, because the
second general problem is only likely to be solved as being
the inverse of the first. All mechanics therefore relate essentially to
the combination of forces, either because their co-operation results in a
movement of which the various circumstances must be studied, or because by
their mutual neutralization the body is in a state
of equilibrium in question. to set the characteristic conditions.
The two general problems, one direct, the other inverse, in the
solution of which the science of mechanics consists, have, under the
report of applications, equal importance; for sometimes
simple motions can be immediately studied by observation,
while the knowledge of the motion which will result from their
combination can only be obtained by theory; and sometimes, on the
contrary, compound motion alone can be effectively observed,
while simple motions, of which we will regard it as the product,
are capable of being determined only rationally. Thus, for
example, in the case of the oblique fall of heavy bodies to the surface
of the earth, we know the two simple movements which the
body would take by the isolated action of each of the forces by which it is animated.
to know, the direction and the speed of the uniform movement which would produce
the only impulse, and the law of acceleration of the
varied vertical movement , which would result from the only gravity; therefore, we propose to
discover the various circumstances of the compound movement produced by
the combined action of these two forces, that is to say to determine the
trajectory that the mobile will describe, its direction and its speed acquired at
each instant, the time it will take to reach a certain
position, etc .; we can, for more generality, add to the two
given forces the resistance of the surrounding environment, provided that the law
is also known. Celestial mechanics presents a capital example
of the reverse question, in determining the forces which produce
the motion of the planets around the sun, or of the satellites around the
planets. One can then know immediately only the
compound movement , and it is according to the characteristic circumstances of this
movement, such as Kepler's laws have summarized them, that it is necessary to
go back to the elementary forces of which the stars must be conceived
animated. to correspond to the effective movements; Once these forces are
known, geometers can usefully take up the question from the
opposite point of view, which it would have been impossible to follow originally.
The true general destination of rational mechanics being
thus clearly conceived,
let us now consider the fundamental principles on which it rests, and first let us examine a
philosophical artifice of the greatest importance in relation to the way in which
bodies are to be considered in mechanics. This conception deserves
all the more our attention since it is still usually
surrounded by a thick metaphysical cloud, which makes it misunderstand its true
nature.
It would be entirely impossible to establish any general proposition
on the abstract laws of equilibrium or of motion, if we did not
begin by looking at bodies as absolutely _inert_,
that is to say as completely incapable of modifying spontaneously
the action of the forces applied to them. But the way this
basic conception is ordinarily presented
strikes me as radically flawed. First of all this abstract notion, which is only a
simple logical artifice imagined by the human mind to facilitate the
formation of rational mechanics, or rather to make it
possible, is often confused with what is very improperly called
_the law of inertia_, which must be regarded, as we will see
below, as a general result of observation. In the second place, the
character of this idea is usually so indecisive, that we do not know
exactly whether this passive state of bodies is purely hypothetical.
or if it represents the reality of natural phenomena. Finally, it
frequently results from this indeterminacy that the mind is unwittingly
inclined to regard the general laws of rational mechanics as
being by themselves exclusively applicable to what we call
gross bodies, while they necessarily hold true, on the
contrary, just as well in organized bodies, although their
precise application encounters much greater difficulties there. It is
very important to correct the
usual notions in these various respects .
We must clearly recognize above all that this passive state of
bodies is a pure abstraction, directly contrary to their true
constitution.
In the way of philosophizing originally employed by the
human mind , one conceived, in fact, matter as being really by its
nature essentially inert or passive, all activity
necessarily coming to it from without, under the influence of certain supernatural beings
or of certain metaphysical entities. But since
positive philosophy began to prevail, and the human mind has confined itself to
studying the true state of things, without inquiring into the
first and generating _causes_ , it has become evident to any observer
that the various bodies natural ones show us a
more or less extensive spontaneous activity . There is in this respect, between the
raw bodies and those which we call par excellence _animés_, only
simple differences of degree. First of all, the progress of
natural philosophy has fully demonstrated, as we shall see
especially later, that there is no living matter
properly so called _sui generis_, since we find in animate bodies
elements exactly identical to those presented by
inanimate bodies . Moreover, it is easy to recognize in the latter a
spontaneous activity exactly analogous to that of living bodies, but
only less varied. If there were
no other property in all material molecules than gravity, that would be enough to prohibit all
physicist to regard them as essentially passive. It would be in
vain for anyone to want to present bodies from a point of view
entirely inert in the act of gravity, saying that they
then only obey the attraction of the terrestrial globe. Even
if this consideration were correct, we would obviously only have displaced the difficulty,
by transferring to the total mass of the earth the activity refused to
isolated molecules. But, moreover, it is clearly seen that, in its fall
towards the center of our globe, a heavy body is just as active as
the earth itself, since it is proved that each molecule of this body
attracts an equivalent part of the earth as much as it is
attracted, although this last attraction alone produces a
perceptible effect , in proportion to the immense inequality of the two masses. Finally, in
a host of other phenomena equally universal, thermological,
electrical, or chemical, matter evidently presents to us a
very varied spontaneous activity, of which we could no longer conceive it
entirely deprived. Living bodies really offer us in this
respect no other particular characteristic than to manifest, in addition to all these
various kinds of activity, some which are peculiar to them, and which
physiologists moreover tend more and more to consider. as a
simple modification of the precedents. Anyway, it is
incontestable that the purely passive state, in which the bodies are
considered in rational mechanics, presents, under the
physical point of view , a true absurdity.
Let us now examine how it is possible that such a supposition
may be employed without any inconvenience in establishing the
abstract laws of equilibrium and motion, which will none the less be
liable thereafter to be properly applied to real bodies.
It suffices, for this, to have regard to the important preliminary remark
recalled above, that the motions are simply considered in
themselves in rational mechanics, without any regard to the mode.
any of their production. From this obviously results, in order to
conform to the language adopted, the faculty of replacing at will any
force by another of any kind, provided that it is capable
of imparting to the body exactly the same movement. From this
obvious consideration, it is conceivable that it is possible to
disregard the various forces which are really inherent in
bodies, and to regard these as only solicited by
external forces , since we can substitute for these internal forces of
mechanically equivalent external forces. Thus, for example,
although every body is necessarily heavy, and we cannot
even to really conceive of a body which would not be it, the geometers
consider, in abstract mechanics, the bodies as being first
entirely stripped of this property, which is implicitly
included in the number of the external forces, if one considered, as
appropriate, a system of forces is entirely arbitrary. Whether the body,
in its fall, is moved by an internal attraction, or whether it obeys
a simple external impulse, is irrelevant for
rational mechanics , if the effective movement is found to be exactly
identical, and we could by therefore preferably adopt the
latter design. This is necessarily so in relation to
any other natural property, which it will always be possible to
replace by the supposition of an external action, constructed in such a way as
to produce the same movement, which will allow the
body to be represented as purely passive; only, as observation or
experience make the laws of these
internal forces known more precisely , it will always be necessary to modify accordingly the
system of external forces which are hypothetically substituted for them,
which will often lead to a very -great complication. Thus, for example,
observation having learned that the vertical motion of a body by virtue
of its gravity is not uniform, but continuously accelerated, we
will not be able to assimilate it to that which a
single impulse would impart to the body , the action of which would not be renewed, since this
would obviously result in a constant speed: we will therefore be obliged to
conceive of the body as having received successively, at
infinitely small time intervals , an infinite series of infinitely small shocks,
such that, the speed produced by each being added
continuously to that which results from the set of precedents, the
effective movement is indefinitely varied; and if the experiment proves that
the acceleration of the movement is uniform, we will suppose all these
successive shocks constantly equal to each other: in any other case, we will need
to suppose them, either for the direction, or for the intensity, a
relation exactly conforming to the real law of the variation of the
movement; but, under these conditions, it is clear that substitution will
always be possible.
It would be useless to insist much to make felt the indispensable
necessity of supposing the bodies in this completely passive state, where
one only has to consider only the external forces which are
applied to them, in order to establish the abstract laws. balance and
movement. It is understandable that if it were necessary first to take into account
any modification which the body may make, by virtue of its
natural forces, in the
one could not establish, in rational mechanics, the slightest
general proposition , all the more so as this modification is far, in most
cases, from being exactly known. It is therefore only by starting by
completely disregarding it, to think only of the reaction of forces
on each other, that it becomes possible to found an
abstract mechanics , from which we will then move on to mechanics. concrete, by
restoring to the bodies their natural active properties, originally
discarded. This restitution constitutes, in fact, the main
difficulty one experiences in making the transition from the abstract to the
concrete in mechanics, a difficulty which singularly limits
reality the important applications of this science, whose
theoretical domain is, in itself, necessarily indefinite. In order to give an
idea of the scope of this fundamental obstacle, we can say that, in
the present state of mathematical science, there is really only one
natural and general property of bodies which we know how to take into
account. 'a suitable way is gravity, either terrestrial
or universal; and still it is necessary to suppose, in this last case, that
the form of the bodies is sufficiently simple. But if this property is
complicated by some other physical circumstances, such as the
resistance of the media, friction, etc., if even bodies are
only supposed to be in the fluid state, it is still only very
imperfectly that we have succeeded so far in appreciating its influence
in mechanical phenomena. A fortiori is it
impossible for us to take into consideration the electrical or
chemical properties, and, still less still, the physiological properties. So
the great applications of rational mechanics have
so far been really limited to celestial phenomena alone, and even to
those of our solar system, where it suffices to have only regard to
a general gravitation, the law of which is simple and well. determined, and
which nevertheless presents difficulties that we do not yet know
overcome completely, when we want to take an exact account of all
the side actions likely to have appreciable effects. We can thus see
to what degree the questions must become more complicated when we pass to
terrestrial mechanics, of which most of the phenomena, even the
simplest, will probably never involve, given the weakness of our
real means, a purely rational study and yet exact according to
the general laws of abstract mechanics, although the knowledge of
these laws, moreover obviously indispensable, can often lead to
important _indications_.
After explaining the true nature of the fundamental design
relative to the state in which bodies are to be assumed in
rational mechanics, it remains for us to consider the general facts or
the _physical laws of motion_ which can furnish a real basis
for the theories of which science is composed. This important exposition is
all the more essential, since, as I indicated above,
since we deviated from the route followed by Newton, we have
completely misunderstood the true character of these laws, whose
ordinary notion is still essentially metaphysical.
The fundamental laws of motion seem to me to be able to be reduced to
three, which must be considered as simple results of
observation, of which it is absurd to want to establish _a priori_ the
reality, although it has been attempted frequently.
The first law is that which is very wrongly designated under the name of
_law of inertia_. It was discovered by Kepler. It consists
properly in that all movement is naturally rectilinear and
uniform, that is to say that any body subjected to the action of
any single force whatever, which acts on it instantaneously, moves constantly
in a straight line and with an invariable speed. The influence of the
metaphysical mind is particularly evident in the way this
law is commonly presented. Instead of just looking at her as
an observed fact, it has been claimed to demonstrate abstractly, by an
application of the principle of sufficient reason, which has not the slightest
solidity. Indeed, to explain, for example, the necessity of
the rectilinear movement, it is said that the body had to follow the straight line,
because there is no reason why it deviates from one side rather than
another from its original direction. It is easy to see
the radical invalidity and even the complete insignificance of such an
argument. First, how could we be sure _that there is
no reason_ for the body to deviate? what can we know in this
regard, other than by experience? The _a priori_ considerations
founded on the _nature_ of things, are they not completely
and necessarily forbidden to us in positive philosophy? Besides, such a
principle, even if one admits it, in itself entails only a
vague and arbitrary application. For, at the origin of the movement,
that is to say at the very moment when the argument should be used, it is
clear that the trajectory of the body does not yet have a
determined geometric character , and that it It is only after the body has
covered a certain space that we can see which line it describes.
It is obvious, by geometry, that the initial movement, instead
of being regarded as rectilinear, could be indifferently supposed
circular, parabolic, or following any other line tangent to the
effective trajectory, so that the same argumentation repeated for
each of these lines, which would be just as legitimate, would lead to
an absolutely indeterminate conclusion. If we reflect on
such reasoning, we will soon recognize that, like all
so-called metaphysical explanations, it really
boils down to repeating in abstract terms the fact itself, and to saying that bodies
have a tendency natural to move in a straight line, which was
precisely the proposition to be established. The insignificance of these
vague and arbitrary considerations will eventually become palpable if one
remark that, as a result of such arguments, the philosophers of
antiquity, and particularly Aristotle, had, on the contrary,
regarded circular motion as natural to the stars, in that it
is the most _perfect_ of all, a conception which is not is also only
the abstract enunciation of a poorly analyzed phenomenon.
I have confined myself to indicating the criticism of ordinary reasonings
relative to the first part of the law of inertia. We can make
perfectly analogous remarks about the second part, which
concerns the invariability of speed, and which we also claim to be able to
demonstrate abstractly, by limiting ourselves to saying that there is no reason.
so that the body never moves slower or faster than at
the origin of the movement.
It is therefore not on such considerations that we can firmly
establish such an important law, which is one of the necessary foundations
of all rational mechanics. It can only have reality as long as it
is conceived as based on observation. But, from this
point of view, its exactitude is evident from the most
common facts . We have continual opportunity to recognize that a body
of a single force constantly moves in a straight line; and, if
it deviates, we can easily see that this modification is due to
the simultaneous action of some other force, active or passive: finally the
curvilinear motions themselves show us clearly, by the
various phenomena due to what is called the _centrifugal force_, that
bodies constantly preserve their natural tendency to move. in
a straight line. There is hardly any phenomenon in nature
which cannot furnish us with a sensible verification of this law, on
which in part the whole economy of the universe is based. It is the
same with regard to the uniformity of movement. All the facts
prove to us that, if the movement originally imprinted always slows down
gradually and ends by being entirely extinguished, it arises from
resistances which bodies constantly encounter, and without which
experience leads us to think that the speed would remain
constant indefinitely , since we see the duration of this
movement appreciably increase as we decrease the intensity of these obstacles. We
know that the simple movement of a pendulum deviated from the vertical, which,
under ordinary circumstances, is hardly maintained for a few
minutes, could have been prolonged up to more than thirty hours, while reducing as
much as possible the friction at the point of the suspension, and causing
the body to oscillate in a very close vacuum, during
Borda's experiments at the Paris Observatory to determine the length of the pendulum at
seconds relative to the meter. Geometers also cite with great
reason, as a manifest proof of the natural tendency of bodies
to retain their acquired speed indefinitely, the
rigorous invariability which one so clearly observes in celestial movements,
which, being executed in a medium of 'extremely rare, are found in
the circumstances most favorable to a perfect observation of the
law of inertia, and which, in fact, for twenty centuries that they have been studied
with some exactitude, have not yet presented us with the slightest
alteration. certain, as to the duration of rotations, or that of
revolutions, although the sequence of times and the improvement of our
means of appreciation should probably reveal to us
some still unknown variations one day .
We must therefore regard as a great law of nature this
spontaneous tendency of all bodies to move in a straight line and
with constant speed. Considering the extreme confusion of common ideas
relative to this first fundamental principle, it may be useful to
point out expressly here that this natural law is just as
applicable to living bodies as to inert bodies for which it
is often believed to be exclusively established. Whatever the origin of
the impulse it has received, a living body tends to persist, like a
inert body, in the direction of its movement, and to maintain its
acquired speed: only forces
capable of modifying or suppressing this movement can develop in it , while, for
other bodies, these modifications are exclusively due to
external agents . But, even in this case, we can acquire a
direct and personal proof of the universality of the law of inertia, by
considering the very sensible effort which we are obliged to make to
change the direction or the speed of our movement. effective, to such a
point, that when this movement is very rapid, it is impossible
for us to modify or suspend it at the precise instant when we would like it
.
The second fundamental law of motion is due to Newton. It consists
in the principle of constant and necessary equality between action and
reaction; that is, whenever a body is moved by
another in any way, it exerts on it, in the opposite direction,
such a reaction that the second loses, due to the masses, a quantity
of movement exactly equal to that which the first received. We have
sometimes tried to establish also _a priori_, this general theorem of
natural philosophy, which is no more susceptible of it than the
preceding one. But it has been much less the subject of
sophistical considerations , and almost all surveyors now agree in the
to consider, according to Newton, as a simple result of observation,
which here dispenses me from any discussion analogous to that of the law
of inertia. This equality in the reciprocal action of bodies manifests itself
in all natural phenomena, whether bodies act on each
other by impulse, or when they act by attraction; it
would be superfluous to cite examples here. We even have so much
opportunity to note this mutuality in our most
common observations , that we could no longer conceive of a body acting on
another, without the latter reacting on it.
I think I should only indicate, from this moment, concerning this
second law of motion, a remark which seems important to me, and which
moreover will be suitably developed in the seventeenth lesson.
It consists in that the famous principle of d'Alembert, according to
which one manages to transform so successfully all questions of
dynamics into simple questions of statics, is really nothing other
than the complete generalization of Newton's law, extended to
any system of forces. This principle obviously coincides with
that of equality between action and reaction, when only
two forces are considered . Such a correlation now makes it possible to conceive
of d'Alembert's general proposition as having an experimental basis,
while it is so far generally established only on
unsatisfactory abstract considerations.
The third fundamental law of movement seems to me to consist in what
I propose to call the principle of the independence or of the
coexistence of movements, which immediately leads to what is
commonly called the composition of forces. Galileo is, properly
speaking, the real inventor of this law, although he did not
conceive it precisely in the form which I believe I should prefer here.
Considered from the simplest point of view, it boils down to the
general fact that every movement exactly common to all the bodies of a
Any system does not alter the particular movements of these
different bodies with respect to one another, movements which continue to be
executed as if the whole of the system were motionless. To state
this important principle with rigorous precision, which no longer requires
any restriction, it is necessary to conceive that all the points of the system
describe at the same time parallel and equal lines, and to consider that
this general movement, with some speed and in some direction
that it may take place will in no way affect the relative movements.
It would be in vain if we attempted to establish by any idea _a priori_
this great fundamental law, which is no more susceptible of it than the
two previous ones. We could, at the most, conceive that if the bodies
of the system are between themselves in a state of rest, this common displacement, which
obviously does not change either their distances or their respective situations,
could not alter this relative immobility: even again, the
absolute ignorance in which we are necessarily of the intimate nature of bodies and
phenomena, does not allow us to affirm rationally, with
perfect security, that the introduction of this new circumstance will
not modify in an unknown manner the conditions
system primitives . But the insufficiency of such an argument becomes especially
noticeable when one tries to apply it to the most extensive and most
important, to that where the different bodies of the system are in motion
with respect to each other. By endeavoring to disregard, as
completely as possible, the observations so known and so varied which
then make us recognize the physical correctness of this principle, it
will be easy to see that no rational consideration
gives us the right to conclude _a priori_ that the general movement will not cause
any change in the particular movements. This is
so true that when Galileo first exposed this
great law of nature, a host
of _a priori_ objections arose from all sides tending to prove the rational impossibility
of such a proposition, which has not been unanimously admitted, until we have
abandoned the logical point of view in order to place ourselves at the
physical point of view .
It is therefore only as a simple general result of observation
and experience that this law can indeed be firmly established.
But, thus considered, it is evident that no proposition of
natural philosophy is founded on observations so simple,
so diverse, so multiplied, so easy to verify. There is not
a single dynamic phenomenon operating in the real world which can
not offer a tangible proof of it; and the whole economy of the universe
would obviously be fundamentally upset,
this law no longer existed. It is thus, for example, that in the
general motion of a vessel, however rapid it may be and
in whatever direction it takes place, the relative motions
continue to be executed, except for the alterations arising from the roll. and
pitch, just as if the ship were stationary, composing
with full motion for an observer not participating in it.
Likewise, we continually see the general displacement of a
chemical focus , or of a living body, in no way affecting the
internal motions which take place there. It is thus above all, to cite
the most important example, that the movement of the terrestrial globe
in no way disturbs the mechanical phenomena that operate on its surface
or in its interior. We know that the ignorance of this third law of
motion was precisely the main scientific obstacle which
for so long opposed the establishment of
Copernicus' theory , against which such a consideration presented then, in
effect, insurmountable objections, from which the Copernicians had
tried to extricate themselves only by vain metaphysical subtleties before the
discovery of Galileo. But since the movement of the earth has been
universally recognized, geometers have rightly presented it
as offering itself an essential confirmation of the reality of
this law. Laplace has proposed a very
ingenious indirect consideration on this subject , which I believe useful to indicate here, because it shows us
the principle of the independence of movements under the verification of a
continuous and very sensitive experience. It consists in observing that,
if the general motion of the earth could in any way alter
the particular motions which are executed on its surface, this
alteration could obviously not be the same for all these motions
whatever their direction, and that they would necessarily be
differently affected according to the greater or lesser angle that
this direction would make with that of the movement of the globe. So, for example, the
oscillatory motion of a pendulum should then present to us
very considerable differences according to the azimuth of the vertical plane in
which it is executed, and which gives it a direction sometimes in conformity,
sometimes in the opposite, and very unequally opposite, to that of the movement. of
the earth; while experience never shows us
the slightest variation in this respect, even when measuring the phenomenon with the extreme
precision which in this respect involves the present state of our means
of observation.
In order to prevent any inaccurate interpretation and any
vicious application of the third law of motion, it is important to note that,
by its nature, it is only relative
that it should never be extended to any rotational movement. The
movements of translation are evidently, in fact, the only ones which
can be rigorously common, in degree as well as in
direction, to all the various parts of any system whatever.
This exact parity can never take place when it is a question of a
movement of rotation, which always necessarily presents
inequalities between the various parts of the system, according to whether they are
more or less distant from the center of the rotation. This is why any
movement of this kind constantly tends to alter the state of the system, and
indeed alters it if the conditions of connection between the various
parts do not provide sufficient strength. Thus, for
example, in the case of a vessel, it is not the general movement of
progression which can disturb the particular movements; the disturbance
is only due to the secondary effects of rolling and pitching, which are
rotational motions. That a watch is simply transported in
any direction with as much speed as one likes, but without
turning at all, it will never be affected; while a mediocre
movement of rotation will suffice alone to promptly disturb its progress.
The difference between these two effects would become especially noticeable, by
repeating the experiment on a living body. Finally, it is as a result of
such a distinction, that we could not have any means of ascertaining,
by purely terrestrial phenomena, the reality of the
translational movement of the earth, which could only be discovered by
celestial observations; while, relatively to its movement of
rotation, it necessarily determines on the surface of the earth, considering
the inequality of centrifugal force between the different points of the globe,
very sensitive phenomena, although not very considerable, the analysis of which
might suffice. to demonstrate, independently of any
astronomical consideration , the existence of this rotation.
The principle of the independence or of the coexistence of movements being
once established, it is easy to conceive that it leads immediately
to the elementary rule ordinarily used for what is called the
_composition of forces_, which is really nothing but a new
way of considering and stating the third law of motion. In
fact, the proposition of the parallelogram of forces, considered
from the most positive point of view, properly consists in that, when a
body is animated at the same time by two uniform motions in
any directions whatever, it describes, by virtue of their combination, the
diagonal of the parallelogram of which he would have at the same time
separately described the sides by virtue of each isolated movement. Or is not it
there evidently a simple direct application of the principle of
the independence of movements, according to which the particular movement of the
body along a certain straight line is in no way disturbed by the
general movement which carries along parallel to itself the totality of the body.
this line along any other straight line? This consideration
immediately leads to the geometric construction stated by the rule
of the parallelogram of forces. This is how this fundamental theorem
of rational mechanics seems to me to be presented directly as
a natural law, or at least as an immediate application of one of
the greatest laws of nature. This is, in my opinion, the only way
truly philosophical to establish this important
proposition solidly , to definitively remove all the metaphysical clouds with
which it is still surrounded and to completely shield it from
any real objection. All the so-called analytical demonstrations
that we have successively tried to give from
purely abstract considerations , besides being usually based on a
vicious interpretation and on a false application of the
analytical principle of homogeneity, moreover suppose that the proposition is
_evidente_ by itself in certain particular cases, when the two
forces, for example, act along the same straight line, evidence which does not
can result then only from the effective observance of the natural law
of the independence of the movements, the indispensableness of which is thus found
indisputably manifested. It would be strange, indeed, for anyone who
considers the question directly from a philosophical point of view,
if, by simple logical combinations, the human mind could thus
discover a real law of nature, without consulting the
outside world in any way .
This notion being of the utmost importance as to the way of
conceiving rational mechanics, and departing greatly from the course
usually adopted today, I believe I should present it again.
under a last point of view which will complete the clarification, by showing
that, despite all the efforts of geometers to elude in this respect
the use of experimental considerations, the physical law of
the independence of movements remains implicitly, even of their
unanimous admission , one of the essential bases of mechanics, although presented
in a different form and at a different time of the exhibition.
It suffices, for that, to recognize that this law, instead of being
exposed directly in the study of the prolegomena of science, is
found later accepted by all the geometers, as establishing the
principle of the proportionality of the speeds to the forces. , necessary basis
of ordinary dynamics.
In order to properly grasp the true character of this question, it
should be noted that the relations of forces can be determined in
two different ways, either by the static process or by the
dynamic process. In fact, we do not always judge the relation of
two forces according to the greater or lesser intensity of the movements which
they can impart to the same body. We
also frequently appreciate it from simple considerations of mutual equilibrium, by considering
as equal the forces which, applied in the opposite direction, following the
same straight line, mutually destroy each other, and then as double,
triple, etc. on the other, the force that would balance two, three,
etc., forces equal to this one, and all directly opposed to the
second. This new means of measurement is, in reality, just as common
as the previous one. That being said, the question essentially consists in
knowing whether the two means are always and necessarily equivalent,
that is to say if, the relations of forces being initially only defined
by the static consideration, it will follow, under the
dynamic point of view , that they will impart to the same mass speeds which
are exactly proportional to them. This correlation is by no means
self-evident; at the most, one can conceive _a priori_ that
the greatest forces must necessarily give the greatest
speeds. But observation alone can decide whether it is to the first
power of the force or to any other increasing function that the
speed is proportional.
It is in order to determine what is, in this respect, the true law of
nature, that, according to all geometers, and particularly of
Laplace, we must consider the general fact of the independence or of the
coexistence of movements. . It is easy to see,
from Laplace's reasoning, that the theory of the proportionality of
velocities to forces is a necessary and immediate consequence of this
general fact, applied to two forces acting in the same
direction. For if a body, by virtue of a certain force, has traveled
a space determined along a certain straight line, and that one comes to
add, in the same direction, a second force equal to the first;
according to the law of the independence of movements, this new force
will only displace the whole of the line of application by an equal
quantity at the same time, without altering the movement of the body along
this line, in so that by the composition of the two movements, this
body will have effectively traversed a space double that which
corresponded to the primitive force. This is the only way in which we
can really ascertain the general proportionality of velocities
to forces, which I must thus dispense with considering as a
fourth fundamental law of motion, since it comes within the
third.
It is therefore evident that, when we thought we could dispense in
mechanics from the general fact of the independence of movements to establish
the fundamental law of the composition of forces, the need to
regard this proposition of natural philosophy as one of the
indispensable bases of science has inevitably reproduced itself to
demonstrate the no less important law of forces proportional to
velocities, which puts this necessity beyond all dispute. So
what was the real result of all the intellectual efforts that have
been attempted to avoid directly introducing into the prolegomena
of mechanics, this fundamental observation? only to appear
to dispense with it in statics, and of course to take it into
consideration only as soon as one passes to the dynamic. Everything is
therefore effectively reduced to a simple transposition. It is clear that such an insignificant
result is in no way proportionate to the
complication of the indirect methods which have been employed to
achieve it, even though these methods would be logically irreproachable,
and we have expressly acknowledged the contrary. It is therefore, in all
respects, much more satisfying to conform frankly and
directly to the philosophical necessity of science, and, since it
It is impossible to do without an experimental basis, to clearly recognize
this basis from the start. No other course can make
completely positive a science which, without such foundations,
would still retain a certain metaphysical character.
These, then, are the three physical laws of motion which provide
rational mechanics with a sufficient experimental basis, on which
the human mind, by simple logical operations, and without
further consulting the outside world, can firmly establish the
systematic edifice of science. Although these three laws seem to me to be
sufficient, I see _a priori_ no reason not to increase the
number, if we could indeed see that they are not
strictly complete. This increase would appear to me to be a very slight
inconvenience for the rational perfection of science, since these
laws could obviously never be greatly multiplied; I
would consider it preferable, as a general thesis, to establish one or
two more, if, in order to avoid it, we had to have recourse to
considerations too diverted, which were of a nature to alter the
positive character of science. But the set of the three laws
above exposed satisfactorily fulfills, in my eyes, all the
essential conditions really imposed by the nature of the theories
of rational mechanics. Indeed, the first, that of Kepler,
completely determines the effect produced by a single force acting
instantaneously: the second, that of Newton, establishes the
fundamental rule for the communication of movement by the action of bodies
on each other. ; finally the third, that of Galileo, leads
immediately to the general theorem relative to the composition of
movements. It is conceivable, from this, that all the mechanics of
uniform motions or instantaneous forces can be entirely
treated as a direct consequence of the combination of these three
laws, which, being by their nature extremely precise, are evidently
capable of being immediately expressed by
easily obtainable analytical equations . As for the most extensive and
important part of mechanics, that which essentially constitutes its
difficulty, that is to say the mechanics of varied motions or
continuous forces , one can conceive, in a general way, the possibility of
reducing it to elementary mechanics, the
character of which we have just indicated , by applying the infinitesimal method, which will make it possible
to substitute, for each infinitely small instant, a
uniform movement for the varied movement, from which will immediately result the
differential equations relating to this last kind of
movements. It will undoubtedly be very important to establish directly and
precisely in the following lessons the general manner
of employing such a method to solve the two essential problems
of rational mechanics, and to carefully consider the
principal results which the surveyors have. thus obtained relative
to the abstract laws of equilibrium and movement. But it is, from this
moment, evident that science is really founded by
the whole of the three physical laws established above, and that all the
work henceforth becomes purely rational, having to consist only
in the use to be made of these laws for the solution of the different
general questions. In short, the separation between the
necessarily physical part and the simply logical part of science
seems to me to be able to be thus clearly effected in an exact and
definitive manner.
To complete this general overview of the philosophical character of
rational mechanics, it only remains for us now to
briefly consider the main divisions of this science, the
secondary divisions to be considered in the following lessons.
The first and most important natural division of mechanics
consists in distinguishing two orders of questions, according to whether one proposes
the search for the conditions of equilibrium, or the study of the laws of
movement, hence the _statique_, and the _dynamique_. It suffices to indicate
such a division, to make its
general necessity directly understood . Besides the effective difference which obviously exists between these
two fundamental classes of problems, it is easy to conceive _a
priori_ that questions of statics must be, in general, by their
nature, much easier to deal with than questions of dynamics.
This results essentially from what, in the former, one does,
as has been rightly said, _abstraction of time_; that is to say that,
the phenomenon to be studied being necessarily instantaneous, one does not
need to have regard to the variations that the forces of the system can
to experience in the various successive instans. This consideration which
, on the contrary, must be introduced into any question of dynamics,
constitutes an additional fundamental element, which makes it the main
difficulty. It follows, as a general thesis, from this radical difference,
that the whole of statics, when it is treated as a
particular case of dynamics, corresponds only to the
much simpler part of the dynamic, to that which concerns the theory.
uniform movements, as we will especially establish in the
next lesson.
The importance of this division is clearly verified by
the general history of the effective development of the human mind. We
let us see, in fact, that the ancients had acquired some
very essential fundamental knowledge relative to equilibrium, either of
solids or of fluids, as can be seen above all by the fine
researches of Archimedes, although they were still very distant to
have a truly complete rational statics. On the contrary, they
completely ignored even the most basic dynamics; the
first creation of this very modern science is due to Galileo.
After this fundamental division, the most important distinction to be
established in mechanics consists in separating, either in statics or
in dynamics, the study of solids and that of fluids. Some
essential that this division is, I place it only in the second line,
and subordinate to the preceding one, according to the method established by Lagrange,
because it is, it seems to me, to exaggerate its influence to constitute it
principal division, as is still done in ordinary treatises
on mechanics. The essential principles of statics or dynamics are,
in fact, necessarily the same for fluids as for
solids; only the fluids require to add to the
characteristic conditions of the system one more consideration, that relating to
the variability of form, which generally defines their
own mechanical constitution . But, while placing this distinction in the rank
appropriate, it is easy to conceive _a priori_ its extreme
importance, and to feel, in general, how much it must increase the
fundamental difficulty of questions, either in statics, or
especially in dynamics. For this perfect reciprocal independence
of molecules, which characterizes fluids, makes it necessary to consider
each molecule separately, and, consequently, to always consider,
even in the simplest case, a system composed of an infinity of
distinct forces. The result, for statics, is the introduction of a
new order of research, relative to the figure of the system in
the state of equilibrium, a very difficult question by its nature, and whose
general solution is still little advanced, even for the sole case of
universal gravity. But the difficulty is even more noticeable in
the dynamics. In fact, the obligation in which we then find ourselves strictly
to consider the specific motion of each molecule separately, in order to make
a truly complete study of the phenomenon, introduced into the question,
considered from the analytical point of view, a complication up to 'at
present inextricable in general, and which one has not yet been able to
overcome, even in the very simple case of a fluid only moved by its
terrestrial gravity, only with the help of very precarious hypotheses, such as
that of Daniel Bernouilli on the parallelism of the slices, which
significantly alter the reality of phenomena. We can
therefore conceive , in general thesis, the greatest necessary difficulty of
hydrostatics, and especially of hydrodynamics, compared to
statics and dynamics proper, which are indeed much more
advanced.
We must add to what precedes, in order to get a fair general idea
of this fundamental difference, that the characteristic definition by
which geometers distinguish between solids and fluids in
rational mechanics, is not really, with regard to some as with
regard to others, that an exaggerated representation, and, consequently,
strictly unfaithful of reality. Indeed, with regard to fluids
mainly, it is clear that their molecules are not
really in that rigorous state of mutual independence in which we
are obliged to assume them in mechanics, by
only subjecting them to maintaining a constant volume between them if it is a question of a
liquid, or, if it is a gas, a variable volume according to a
given function of the pressure, for example, in inverse ratio to this
pressure, according to Mariotte's law. A large number of
natural phenomena are, on the contrary, essentially due to the mutual adhesion
of the molecules of a fluid, a bond which is only much less
than in solids. This membership, which is ignored for
mathematical fluids, and which it seems, in fact, almost impossible to
take properly into consideration, determines, as we know,
very sensible differences between the effective phenomena and those which
result from the theory, either for statics or above all, for the
dynamics, for example relative to the flow of a heavy liquid
through a determined orifice, where the observation deviates notably from the
theory as to the expenditure of liquid in a given time.
Although the mathematical definition of solids is found to represent
their real state much more accurately, there are, however, several
occasions to recognize the need to take account in certain cases of
the possibility of mutual separation which always exists between the
molecules of a solid, if the forces which are applied to them
acquire a sufficient intensity, and which one completely
disregards in rational mechanics. This can easily be
seen especially in the theory of the rupture of solids, which,
hardly sketched out by Galileo, by Huyghens, and by Leïbnitz, is
today in a very imperfect and even very precarious state. , despite
the work of several other surveyors, and which would nevertheless be
important to shed light on several questions of terrestrial mechanics,
mainly of industrial mechanics. It should be noted, however,
this subject, that this imperfection is at the same time much less sensitive
and much less important than that noted above, relative to the
mechanics of fluids. For it is found to be in no way able to influence
questions of celestial mechanics, which really constitute,
as we have had several occasions to recognize, the main
application, and probably the only one which can ever be truly
complete, of rational mechanics.
Finally we must also point out, in general thesis, in
current mechanics , a lacuna, secondary it is true, but which is not without
importance, relative to the theory of a class of bodies which are
in an intermediate state between rigorous solidity and fluidity,
and which could be called semi-fluids, or semi-solids: such are for
example, on the one hand, sands, and, on the other hand, fluids in
the gelatinous state. Some
rational considerations have been presented about these bodies, under the name of imperfect fluids,
especially with respect to their equilibrium surfaces. But their
own theory has never really been established in a general and
direct manner.
These are the main general insights that I thought it necessary to indicate
summarily in order to appreciate the philosophical character which
distinguishes rational mechanics, considered as a whole. he
It is now a question, by considering
the effective composition of science from the same philosophical point of view , of appreciating
how, by the important successive works of the greatest
geometers, this second general section, so extensive, so essential, and
so difficult to concrete mathematics, has been able to be raised to that eminent
degree of theoretical perfection which it has attained in our days in
the admirable treatise of Lagrange, and which presents to us all the
abstract questions which it is capable of offering, brought back, according to
a single principle, to depend only on purely
analytical research , as we have already recognized for the problems
geometric. This will be the subject of the following three lessons; the first
devoted to _statics_, the second to dynamics, and the third to
the examination of general theorems of rational mechanics.
SIXTEENTH LESSON.
SUMMARY. General view of the statics.
The whole of rational mechanics can be treated according to two
general methods which are essentially distinct and unequally perfect,
according to whether statics is conceived in a direct manner, or whether it is
considered as a special case of dynamics. By the first
method, we immediately take care of discovering a
sufficiently general principle of equilibrium , which we then apply to the determination of
equilibrium conditions of all possible force systems. By the
second, on the contrary, we first seek what would be the motion
resulting from the simultaneous action of any of the various forces
proposed, and we deduce therefrom the relations which must exist between these
forces for this motion to be zero.
Statics being necessarily of a simpler nature than
dynamics, the first method alone could have been used at the origin of
rational mechanics. It is, in fact, the only one that was known to the
ancients, entirely foreign to any idea of dynamics, even the most
elementary. Archimedes, true founder of statics, and to whom are
due to all the essential notions that antiquity possessed in this
regard, begins to establish the condition of equilibrium of two weights
suspended at the two ends of a right lever, that is to say the
need for these weights to be due inverse of their distances to
the fulcrum of the lever; and he then strives to reduce as much as
possible to this single principle the search for equilibrium relations
specific to other systems of forces. Similarly, with regard to the statics
of fluids, he first poses his famous principle, consisting in that
any body immersed in a fluid loses a part of its weight equal to the
weight of the displaced fluid; and then he deduces, in a large number
case, the theory of the stability of floaters. But the principle
of the lever was not in itself sufficiently general to make
it possible to apply it really to the determination of the
conditions of equilibrium of all systems of forces. By some
ingenious artifices which one successively tried to extend
the use of it, one could effectively bring back to it only the systems composed of
parallel forces. As for the forces whose directions concur, we
first tried to follow a similar course, by imagining new
direct principles of equilibrium specially peculiar to this more
general case , and among which we must especially note the happy idea of
Stévin, relating to the balance of the system of two weights placed on two back-to-back
inclined planes. This new parent idea would perhaps have been sufficient
strictly to fill the gap left in statics
by Archimedes' principle, since Stévin had succeeded in deducing
from it the equilibrium relations between three forces applied at the same point,
in the case of at least where two of these forces are at right angles; and he
had even noticed that the three forces are then between them like
the three sides of a triangle whose angles would be equal to those
formed by these three forces. But, the dynamic having been founded at the
same time by Galileo, the surveyors ceased to follow the old
direct static walk, preferring to search for the
conditions of equilibrium according to the known laws of the
composition of forces. It was by this last method that Varignon
discovered the true general theory of the equilibrium of a system of
forces applied at the same point, and that later d'Alembert
finally established , for the first time, the equilibrium equations of
any system of forces applied to different points of a solid body
of invariable form. This method is still the most
universally used today.
At first glance, it seems not very rational, since, the dynamic
being more complicated than the static,
to make it depend on the other. On
the contrary, if possible, it would be more philosophical to reduce dynamics to
statics, as we have since achieved. But we must nevertheless
recognize that, in order to treat statics completely as a
particular case of dynamics, it suffices to have formed only
the most elementary part of it, the theory of uniform motions,
without having any need for the theory. various movements. It is important
to explain this fundamental distinction precisely.
To this end, let us first observe that there are, in general, two kinds of
forces: 1 ° the forces that I call _instantaneous_, like
impulses, which act on a body only at the origin of movement,
leaving it to itself as soon as it is in motion; 2 ° the forces which
we call rather improperly _accelerator_, and which I prefer to
name _continues_, like the attractions, which act unceasingly on
the mobile during the whole duration of the movement. This distinction
is evidently equivalent to that of _uniform_ movements and
_varied_ movements ; for it is clear, by virtue of the first of the three
fundamental laws of motion exposed in the previous lesson, that any
instantaneous force must necessarily produce a uniform motion,
while any continuous force must, on the contrary, by its nature,
give the mobile an indefinitely varied movement. That said, one
can easily conceive, _a priori_, as I have already indicated several
times, that the part of the dynamics relating to instantaneous forces or
to uniform movements must be, without any comparison, infinitely
simpler than that which concerns the continuous forces or
various motions , and in which essentially consists all the difficulty of
the dynamic. The first part presents such ease that it
can be treated as a whole as an immediate consequence of the
three fundamental laws of motion, as I have expressly
noticed at the end of the previous lesson. Now it is easy to
to conceive, in general thesis, that it is only of this first
part of the dynamics that one needs to constitute the statics as
a particular case of the dynamics.
Indeed, the phenomenon of equilibrium, of which it is then a question of discovering
the laws, is obviously, by its nature, an instantaneous phenomenon, which
must be studied without any regard to time. The consideration of time is
introduced only in research relating to what is called the
_stability_ of equilibrium; but this research is no longer,
strictly speaking, part of the statics, and comes essentially
into the dynamic. In short, following the ordinary aphorism already mentioned,
we always, in statics, disregard time. As a result, we
can regard all the forces that we consider as instantaneous,
without the theories ceasing to have all the
necessary generality . For, at each epoch of its action, a continuous force can
obviously always be replaced by an instantaneous force
mechanically equivalent, that is to say capable of imparting to the moving body
a speed equal to that which actually gives it at that instant
proposed force. In truth, it will be necessary, in the
next infinitely small moment , to substitute for this instantaneous force a new force of the same
nature, to represent the effective change of speed, of such
so that, in dynamics, where one must consider the state of the moving body in
the various successive instans, one will necessarily find by the
variation of these instantaneous forces the fundamental difficulty
inherent in the nature of the continuous forces, and which will not have made that
change shape. But, in statics, where it is only a question of considering the
forces in a single instant, one will not have to take account of these
variations, and the general laws of equilibrium, thus established by
considering all the forces. as instantaneous, will nonetheless be
applicable to continuous forces, provided that care is taken, in this
application, to substitute for each continuous force the instantaneous force
which corresponds to him at the moment.
We can thus clearly see by this how abstract statics can be
treated with facility as a simple application of the most
elementary part of dynamics, that which relates to
uniform motions . The most suitable way to carry out this application
is to notice that, when any forces are in
equilibrium, each of them, considered in isolation, can be
regarded as destroying the effect of the set of all the others.
Thus the search for the conditions of equilibrium is reduced, in general,
to expressing that any one of the forces of the system, is equal and
directly opposed to the _resultant_ of all the others. The
difficulty therefore consists, in this method, only in determining this
resultant, that is to say in _composing_ between them the proposed forces.
This composition is carried out immediately for the case of two forces
according to the third fundamental law of motion, and we
then deduce the composition of any number of forces. The
elementary question presents, as we know, two essentially distinct cases,
depending on whether the two forces to be composed act in
converging directions or in parallel directions. Each of these two cases
can be treated as deriving from the other, from which results among the
geometers a certain divergence in the manner of establishing the
elementary laws of the composition of forces, according to the case
chosen as a starting point. But, without contesting the
rigorous possibility of proceeding otherwise, it seems to me more rational, more
philosophical and more strictly in accordance with the spirit of this manner
of treating statics, to begin with the composition of the forces which
compete, hence the we naturally deduce that of parallel forces
as a particular case, while the inverse deduction can only be made
with the aid of indirect considerations, which, however ingenious
they may be, necessarily present something
forced.
After having established the elementary laws of the composition of forces,
the geometers, before applying them to the search for the conditions of
equilibrium, usually make them undergo an important
transformation, which, without being completely indispensable, presents
nevertheless, under the report analytic, the highest utility, by
the extreme simplification which it introduces into the algebraic expression
of the conditions of equilibrium. This transformation consists in what is
called the theory of _momens_, whose essential property is to
analytically reduce all the laws of the composition of forces to
simple additions and subtractions. The denomination of _momens_,
completely diverted today from its original meaning, now
designates only the abstract consideration of the product of a
force by a distance. We must distinguish, as we know, two kinds
of _momens_, the moments with respect to a point, which indicate the product
of a force by the perpendicular lowered from this point on its
direction, and the moments with respect to a plane, which denote the product
of the force by the distance from its point of application to this plane. The
former obviously depend only on the direction of the force, and in
no way on its point of application; they are especially appropriate
by their nature to the theory of non-parallel forces: the second to
on the contrary, depend only on the point of application of the force, and
not on its direction; they are therefore essentially intended for the
theory of parallel forces. We shall have occasion to indicate below
by what happy fundamental idea M. Poinsot has succeeded in attributing
generally, and in the most natural manner, a
direct concrete meaning to both kinds of moments, which do not had
really only an abstract value before him.
The notion of moments once established, their elementary theory consists
essentially in these two very remarkable general properties,
which can easily be deduced from the composition of forces: 1 ° if we consider
a system of forces all located in the same plane, and
moreover arranged in any way, the moment of their resultant, with
respect to any point of this plane, is equal to the algebraic sum
of the moments of all the components with respect to this same point, by
attributing to these various moments the suitable sign, according to the direction
according to which each force would tend to make its lever arm turn
around the origin of the moments supposedly fixed; 2º considering a
system of parallel forces arranged in any way in
space, the moment of their resultant with respect to any plane
is equal to the algebraic sum of the moments of all the components by
relation to this same plane, the sign of each moment then being
naturally determined, in accordance with ordinary rules, according to the
sign specific to each of the factors of which it is composed. The first of these
two fundamental theorems was discovered by a geometer to whom
rational mechanics owes a lot, and whose memory was worthily
raised by Lagrange from an unjust oblivion, Varignon. The way in which
Varignon establishes this theorem in the case of two components, from which
the general case immediately results, is even especially remarkable.
Indeed, looking at the moment of each force with respect to a point
as obviously proportional to the area of the triangle which would have this point
for vertex and for base the line which represents the force, Varignon,
according to the law of the parallelogram of forces, first presents the
theorem of moments in a very simple geometrical form, by
showing that if, in the plane of a parallelogram, we take
any point , and consider the three triangles having this point
for common vertex, and for bases the two contiguous sides of the
parallelogram and the corresponding diagonal, the triangle constructed
on the diagonal will be constantly equivalent to the sum where
unlike triangles constructed on both sides; which is in
itself, as Lagrange rightly observes, a beautiful theorem of
geometry, regardless of its mechanical utility.
With the help of this theory of moments, we can easily express
the analytical relations which must exist between the forces in
the state of equilibrium, by first considering, for ease of reference, the
two particular cases of a system of forces all located in
any way in the same plane, and of any system of
parallel forces. Each of these two systems requires, in general, three
equilibrium equations, which consist: 1 ° for the first, in that the
algebraic sum of the products of each force, either by the cosine, or
by the sine of the angle qu 'she does with a fixed straight taken
arbitrarily in the plane is separately zero, as well as the
algebraic sum of the moments of all the forces with respect to
any point of this plane; 2 ° for the second, in that the algebraic sum
of all the proposed forces is zero, as well as the algebraic sum
of their moments taken separately with respect to two different planes
parallel to the common direction of these forces. After having treated these
two preliminary cases, it is easy to deduce from them that of a
completely arbitrary system of forces. It suffices, for that, to conceive each
force of the system broken down into two, one located in
any fixed plane , the other perpendicular to this plane. The proposed system is
will therefore be replaced by the set of two
simpler secondary systems , one composed of forces all directed in the same plane,
the other of forces all perpendicular to this plane and consequently
parallel to each other. As these two partial systems
obviously cannot balance each other, it will therefore be necessary, for
equilibrium to take place in the primitive general system, that there
exist in each of them in particular, this which brings the question back to the
two preliminary questions already dealt with. Such is at least the
simplest way of conceiving, by treating statics by the
dynamic method, the general research of the analytical conditions of
equilibrium for any system of forces; although it was
obviously possible, by complicating the solution, to solve
the problem directly in its entirety, so as to include
on the contrary, like a simple application, the two
preliminary cases . Whatever course we consider advisable to adopt, we find
for the equilibrium of any system of forces, the
following six equations : / [SPcos / alpha = 0, /; SPcos / beta = 0, /; SPcos / gamma = 0, /]
/ [SP (ycos / alpha-xcos / beta) = 0, /; SP (zcos / alpha-xcos / gamma) = 0, /]
/ [SP (ycos / gamma-zcos / beta ) = 0; /] denoting by P the intensity of
any one of the forces of the system, by / alpha, / beta, / gamma, the
angles formed by its direction with three fixed rectangular axes
chosen arbitrarily, and by x, y, z, the coordinates of its point
of application relative to these three axes. I use here the
characteristic S to designate the sum of similar products,
specific to all the forces of the system P, P ', P' ', etc.
This is, in essence, the way of proceeding to the determination of the
general conditions of equilibrium, by conceiving statics as a
particular case of elementary dynamics. But, however simple
this method may be, it would obviously be more rational and more
satisfactory to return, if it is possible, to the method of the ancients, in
freeing statics from any dynamic consideration, to proceed
directly to the search for the laws of equilibrium considered in itself,
using a sufficiently general principle of equilibrium, established
immediately. This is indeed what geometers attempted, when
once the general equations of equilibrium were discovered by
the dynamic method. But above all they were determined to establish a
direct static method, by a philosophical motive of a
higher order and at the same time more pressing than the need to present
statics under a more perfect logical point of view. This is now what
it is most important for us to explain, since this is the way
which led Lagrange to imprint on the whole of
rational mechanics that high philosophical perfection which
now characterizes it .
This fundamental motive results from the necessity which one finds to
deal, in general, the most difficult and the most
important questions of the dynamics, to make them return in simple
questions of statics. We will especially examine, in the
following lesson , the famous general principle of dynamics discovered by
d'Alembert, and with the help of which any research relating to the motion
of any body or of any system, can be immediately converted
into a problem. balance. This principle, which, from the point of view
philosophical, is really, as I already indicated in the
previous lesson , only the greatest possible generalization of the second
fundamental law of the movement, have been for nearly a century of
permanent basis for the solution of all the great dynamic problems, and
must obviously now receive more and more such a
destination, given the admirable simplification it brings in
the most difficult searches. Now it is clear that a similar way
of proceeding necessarily obliges to treat statics in its turn by
a direct method, without deducing it from dynamics, which thus is, on the
contrary, entirely based on it. It is not that there is
strictly speaking, no real vicious circle to persist still
in the ordinary course exposed above, since the
elementary part of the dynamics, on which only one made
statics rest , is, in reality, to be completely distinct from that
which one can only treat by reducing it to statics. But it is nonetheless
obvious that the whole of rational mechanics
then presents , by proceeding in this way, only an
unsatisfactory philosophical character , because of the frequent alternative between the
static point of view and the point of view. dynamic view. In short, science, poorly
coordinated, is thereby essentially lacking in unity.
The definitive adoption and universal use of d'Alembert's principle
thus made essential to the future progress of the human mind a
radical overhaul of the entire system of rational mechanics, where, the
statics being treated directly according to a primitive law
sufficiently general equilibrium , and the dynamic recalled to statics,
the whole of science could acquire a character of unity henceforth
irrevocable. Such is the eminently philosophical revolution carried out
by Lagrange in his admirable treatise on _mécanique analytique_,
the fundamental conception of which will always serve as the basis for all
subsequent work of geometers on the laws of balance and motion,
as we have seen the great mother idea of Descartes having to lead
indefinitely all geometrical speculations.
By examining the research of previous geometers on the properties
of equilibrium, in order to draw from it a direct principle of statics which could
offer all the necessary generality, Lagrange stopped to choose
the _principle of virtual speeds_, which has now become so famous by
the immense and capital use he made of it. This principle, first discovered
by Galileo in the case of two forces, as a
general property manifested by the equilibrium of all machines,
had been later extended by Jean Bernouilli to any number
of forces, constituting any system; and Varignon had then
expressly noticed the universal use that it was possible to make of it
in statics. The combination of this principle with that of d'Alembert
led Lagrange to conceive and treat rational mechanics
as a whole as deduced from a single fundamental theorem, and thus to give
it the highest degree of perfection that a science can acquire.
in the philosophical respect, a rigorous unity.
To conceive clearly with more ease the general principle of
virtual speeds, it is still useful to consider it first in
the simple case of two forces, as had done Galileo. It consists
then in that, two forces making equilibrium with the help of
any machine , they are between them in inverse ratio of the spaces which
crossed in the direction of their directions their points
of application, if we suppose that the system came to take an
infinitely small movement: these spaces bear the name of _vitesses sociales_
, in order to distinguish them from the real speeds which would
actually take place if equilibrium did not exist. In this
primitive state , this principle, which one can very easily verify relatively to
all known machines, already presents a great practical utility,
considering the extreme facility with which it makes it possible to obtain effectively
the mathematical condition of equilibrium of any machine, the
constitution of which would even be entirely unknown. By calling _virtual
moment_ or simply _moment_, according to the primitive acceptation of this
term among surveyors, the product of each force by its _virtual speed_
, a product which, in effect, then measures the force of the force
to move the machine, we can greatly simplify the statement of the
principle by limiting himself to saying that, in this case, the moments of the two
forces must be equal and of opposite sign for there to be
equilibrium; the positive or negative sign of each _moment_ is determined
from that of the virtual speed, which will be estimated, in accordance with
the ordinary spirit of the mathematical theory of signs, positive or
negative according to whether, by the fictitious movement which one imagines, the
projection of the point of application would find itself falling on the direction
of the force itself or on its extension. This abbreviated expression of the
principle of virtual speeds is especially useful to state this
principle in a general way, relative to a system of forces
quite any. It then consists in that the algebraic sum
of the virtual moments of all the forces, estimated according to the
preceding rule , must be zero for there to be equilibrium; and this
condition must take place distinctly with respect to all
elementary movements which the system could take by virtue of the
forces by which it is animated. By calling P, P ', P' ', etc., the
proposed forces , and, according to the ordinary notation of Lagrange, / delta / rho,
/ delta / rho', / delta / rho '', etc., the speeds virtual
corresponding ones, this principle is immediately expressed by
the equation / [P / delta / rho + P '/ delta / rho' + P '' / delta / rho '' + / mbox {/ rm
etc.} = 0, /] or, more briefly, / [/ int P / delta / rho = 0, /] in
which, by the work of Lagrange, the whole of rational mechanics
can be regarded as implicitly enclosed. As for
statics, the fundamental difficulty of properly developing this
general equation will be reduced essentially, when all the forces to
be taken into account are well known, to a purely
analytical difficulty , which will consist in relating, in each case, according to the
conditions of connection characteristic of the system considered, all the
variations infinitely small / delta p, / delta p ', etc., to the smallest
possible number of truly independent variations, in order to cancel
separately the various groups of terms relating to each of these
latter variations, which provides, for equilibrium, as many
distinct equations as there could be elementary motions
really different in the nature of the proposed system. Assuming that
the forces are entirely arbitrary, and whether they are applied
to the various points of a solid body, which is not, moreover, subject to
any particular condition, we also immediately and in the
simplest way arrive at the six general equations of l equilibrium
reported above according to the dynamic method. If the solid, instead
of being completely free, must be more or less constrained, it suffices
to introduce into the number of the forces of the system the resistances which
result from them after having suitably defined them, which will
only add a few new ones. terms to the fundamental equation. It is the
same when the shape of the solid is not strictly assumed
invariable, and we come, for example, to consider its elasticity. Such
modifications have no other effect, from a
logical point of view , than to complicate more or less the equation of
virtual speeds , which does not for that cease necessarily to retain its
entire generality, although these secondary conditions can
sometimes make almost inextricable the purely
analytical difficulties presented by the effective solution of the proposed question.
As long as the theorem of virtual speeds had been conceived only as
a general property of equilibrium, we had been able to confine ourselves to
verifying it by its constant conformity with the ordinary laws of
the balance already obtained otherwise, and of which he thus presented a
very useful summary by its simplicity and uniformity. But, in order to make
this fundamental theorem the effective basis of all
rational mechanics , in a word, to convert it into a true principle, it
was essential to establish it directly without deducing it from any
other, or at least by not assuming that preliminary proposals
likely by their extreme simplicity to be presented as
immediate. This is what Lagrange so happily performed by his
ingenious demonstration based on the muffle principle and in
which he generally succeeds in proving the speed theorem
virtual with extreme ease, imagining a single weight, which,
with the help of suitably constructed mittens,
simultaneously replaces all the forces of the system.
Since then, several other direct and general demonstrations of the principle
of virtual speeds have been proposed successively , but which, much more complicated than that
of Lagrange, are, in reality, in no way superior to it as regards
logical rigor. For us, from the philosophical point of view, we
must regard this general theorem as a necessary consequence of
the fundamental laws of motion, from which it can be deduced in various
ways, and which then becomes the effective starting point of the
whole rational mechanics.
The use of such a principle bringing the whole of science to a
perfect unity, it obviously becomes very uninteresting from now on to
know other still more general principles, assuming that we
can obtain some. We can therefore regard as essentially idle
by their nature the attempts that could be planned to
substitute some new principle for that of virtual speeds. One
such work can not be more perfect way the character
fundamental philosophy of rational mechanics, which in the
Treaty of Lagrange, is also highly coordinated as it could ever
being. One could not really have in view any other
effective utility than to considerably simplify the analytical researches to
which science is now reduced, which must seem
almost impossible when one considers with what admirable facility the
principle of virtual speeds has been adapted by Lagrange to
the uniform application of mathematical analysis.
This is therefore the incomparably most perfect way of conceiving
and treating statics, and therefore the whole of
rational mechanics . In a work such as this especially, we could not
hesitate for a single moment to give this method a preference.
brilliant over any other, since its main characteristic advantage
is to perfect the philosophy of this science to the highest degree.
This consideration must have in our eyes much more importance than
we cannot attribute in the opposite direction to the specific difficulties which
it still frequently presents in applications, and which
consist essentially in the extreme intellectual restraint which
it often requires, which may be regarded as being up to a
certain point inherent in any very general method in which
any questions whatever are constantly brought back to a single principle. However,
these difficulties are so far great enough that we cannot
still consider Lagrange's method as really elementary,
so as to be able to dispense entirely from considering any other
in a dogmatic teaching. This is what determined me to
characterize first with a few developments the dynamic method
properly so called, the only one still generally used. But these
considerations can obviously only be provisional; the
main embarrassments caused by the use of Lagrange's conception
having really no other essential cause than its novelty.
No doubt such a method is not intended indefinitely for the exclusive use
of a very small number of surveyors, who alone still have one.
knowledge familiar enough to make proper use of the admirable
properties which characterize it: it was certainly to become
later as popular in the mathematical world as the great
geometrical conception of Descartes, and this general progress would probably
already be almost effected if the fundamental notions of
transcendent analysis were more universally prevalent.
I would not believe that I have adequately characterized all the
essential philosophical notions relating to rational statics, if I
did not now make a separate mention of a
very important new conception , introduced into science by M. Poinsot, and which I
regards it as the greatest improvement that
the general system of mechanics has undergone, from a philosophical point of view, since
the regeneration effected by Lagrange, although it is not exactly
in the same direction. It is a question, as we see, of the ingenious and
luminous theory of couples, which M. Poinsot so happily created
to perfect
rational mechanics directly in his fundamental conceptions , and the scope of which does not seem to me to have been
sufficiently appreciated by most surveyors. We know that
these _couples_, or systems of equal and contrary parallel forces,
had hardly been noticed before M. Poinsot as a sort of
paradox in statics, and that he seized on this isolated notion
to immediately make it the subject of a very extensive and
entirely original theory relating to the transformation, composition and
use of these singular groups, which 'he showed them to be endowed with properties
so remarkable for their generality and their simplicity. These
fundamental properties consist essentially: 1 ° with respect to
direction, in that the effect of a couple depends only on the
direction of its plane or of its axis, and not on the position of this
plane, nor on that of the couple in the plane; 2 ° as regards the intensity, in
that the effect of a couple does not properly depend on the value of each
of the equal forces which compose it, nor of the lever arm on which
they act, but only of the product of this force by this
distance, to which M. Poinsot has rightly given the name of moment of the
couple.
By adopting the dynamic method proper to proceed to the
search for the general conditions of equilibrium, M. Poinsot
presented it from a completely new point of view with the aid of his
conception of couples, which considerably simplified it. and
cleared up. To briefly characterize this variety of the
dynamic method here, it will suffice to conceive that, by adding at
any point of the system two forces equal to each of those which we
considers and which act, in opposite directions from each other, following
a straight line parallel to its direction, we can thus, without ever
altering the state of the proposed system, look at it as
replaced: 1 ° by a system of forces equal to the primitive forces
transported all parallel to their directions at the single point which
one will have chosen, and which, consequently, will be generally
reducible in only one; 2 ° by a system of couples having as a measure
of their intensity the moments of the forces proposed relative to this same
point, and whose planes, all passing through this same point, will
also make them generally reducible to a single couple. We see, according to
that, with what ease one will be able to proceed in this way to the determination
of the equilibrium relations, since it will suffice to find, by the
known laws of the composition of the converging forces, this
unique resultant , in order to express that it is zero; and then, by the laws which
M. Poinsot established for the composition of couples, obtain also
this resulting couple, and also annul it separately; for it is clear
that, since force and couple cannot destroy each other,
equilibrium can only exist by supposing them individually to be
null.
It must undoubtedly be recognized that this new way of proceeding
is not essential in order to apply the dynamic method to
determination of the general conditions of equilibrium. But, in addition to
the extreme simplification that it introduces into such research,
we must above all appreciate, as regards the general progress of science,
the unexpected clarity that it brings to it, that is to say the eminently
lucid aspect. under which it presents an essential part of these
conditions of equilibrium, all those which are relative to the _momens_
of the proposed forces, and which constitute the most important half of the
static equations. These _momens_, which until then indicated only a
purely abstract consideration, artificially introduced into
statics to facilitate the algebraic expression of the laws of equilibrium,
have henceforth taken on a perfectly distinct concrete meaning,
and entered as naturally as the forces themselves into
static speculations, as being the direct measure of the couples to
which these forces immediately give rise. It is easy to see
_a priori_ what facility this general and elementary interpretation
must necessarily provide for the combination of all the ideas
relating to the theory of moments, as we can already see the
effective proof in the extension and improvement of this
important theory, by the work of M. Poinsot himself.
Whatever, in reality, the fundamental qualities of the
M. Poinsot's conception of statics, we must nevertheless
recognize, it seems to me, that it is above all to the improvement of
dynamics that it finds itself, by its nature, essentially destined; and
I think I can assure you, in this respect, that this conception has not
yet exercised its most fundamental influence. It must be
regarded, in fact, as directly suited to perfecting in a
very important respect the very elements of general dynamics, by
rendering the notion of the motions of rotation as natural, as
familiar, and almost as simple as that of the motions of rotation. of
translation. Because the couple can be seen as the natural element
of rotational motion, as well as the force of
translational motion . This is not the place to indicate
this consideration more clearly , which will be suitably reproduced in the
following lessons . We must only conceive, as a general thesis, that a
use of course of the theory of couples establishes the possibility of
rendering the study of rotational motions, which so far constitutes
the most complicated and obscure part of the dynamics. , as
elementary and as clear as the study of the movements of translation.
We will have the opportunity to see later on to what degree
of simplicity and clarity M. Poinsot has succeeded in thus reducing
various essential propositions, relative to the motions of rotation,
and which had been established before him only in the most painful and
indirect manner, principally with regard to the properties of the
_aires_, of which he even appreciably increased the extent and regularized
the application in various important respects, especially, lastly, as
regards the determination of what is called the _invariable plane_.
To complete these philosophical considerations on the whole of
statics, I believe I should add here the summary indication of a
last general notion, which it seems to me useful to introduce into the
theory of equilibrium, in any way that 'we have also judged
suitable to establish it.
When we want to get a fair idea of the nature of the various
equations which express the conditions of the equilibrium of
any system of forces, it is, it seems to me, insufficient to confine ourselves to
noting that the set of these equations is essential for
balance, and inevitably establishes it. It is necessary, moreover, to be able to
assign clearly the static significance distinctly proper to
each of these equations considered in isolation, that is to say to determine
with precision in what each contributes separately to the production of
equilibrium, an analysis to which we cannot is not usually attached,
although it is doubtless important.
proceeds to the establishment of the static equations, it is clear _a
priori_ that the equilibrium can only result from the destruction of all
the elementary movements which the body could take by virtue of the
forces with which it is animated, if these forces had not point between them the
relations necessary to counterbalance exactly. Thus each
equation taken separately must necessarily annihilate one of these motions,
so that the whole of these equations produces equilibrium, by
the impossibility in which the body therefore finds itself from moving in any
way. Let us now briefly examine the general principle according to
which such an analysis seems to me to be capable of
Considering motion from the most positive point of view, as
the simple transport of a body from one place to another, regardless
of whatever mode in which it may be produced, it is evident
that all movement must be considered, in the most general case,
as necessarily composed of both _translation_ and
_rotation_. It is not, without doubt, that there cannot really exist
translation without rotation, or rotation without translation; but we
must regard both cases as being exceptional, the normal case
consisting in fact in the coexistence of these two kinds of movements,
which are constantly accompanied unless there are special conditions.
very precise, and consequently very rare, relative to the circumstances
of the phenomenon. This is so true that the mere verification of one
of these movements is habitually regarded with reason by
geometers, who know the full scope of this
elementary observation , as a powerful motive, not to affirm, but to presume
very much. presumably the existence of the other. Thus, for example, the
only knowledge of the rotational movement of the sun on its axis,
perfectly observed since Galileo, would be _a priori_ for a
geometer an almost certain proof of a translational movement of
this star accompanied by all its planets, all the same. astronomers
would not have already begun to recognize effectively, by
direct observations, the reality of this transport, in a sense as yet
little determined. Likewise, it is from a similar consideration
that we commonly admit, with reason, in addition to the motive of analogy,
the existence of a rotational movement in the planets even in respect of
which we have no we could still see it directly, by that
alone that they have a well-known translational movement around the
sun.
It follows from this first analysis that the equations which express the
conditions of equilibrium of a body, solicited by any forces,
must have as their object,
translation, the others to annihilate any rotational movement. Let us
now see , from the same point of view, in order to complete this
general outline , what must be _a priori_ the number of equations of each
species.
As for the translation, it suffices to consider that, in order to prevent a
body from walking in any direction, it is evidently necessary to
prevent it along three principal axes situated in different planes, and
which we usually suppose to be perpendicular between them. Indeed, what
progression would be possible, for example, in a body which could not
advance neither from east to west or from west to east, nor from north to south
or from south to north, nor finally from top to bottom or from bottom to top? Any
progression in any other direction, which can obviously be
conceived as composed of corresponding partial progressions in
these three main directions, would therefore have become necessarily
impossible. On the other hand, it is clear that one should not consider
less than three independent elementary motions, because the body could
move in the direction of one of the axes, without having any translation
in the direction of either of the two. other. It is thus conceived that, in general,
three equations of condition will be necessary and sufficient to
establish, in any system, the equilibrium of translation; and
each of them will be specially destined to destroy one of the three
elementary movements of translation which the body could take.
We can present an exactly analogous consideration with respect to
rotation: there is no new difficulty than that of
distinctly perceiving a more complicated mechanical image. The rotation of a
body in a plane or around any axis, which can always be considered
broken down into three elementary rotations in the three
coordinated planes or around the three axes, it is clear that, in order to
prevent any rotation in a body, it must also be prevented from
rotating separately with respect to each of these three planes or of these
three axes. Three equations are therefore, similarly, necessary and
sufficient to establish rotational equilibrium; and we see, with
the same facility as in the previous case, the mechanical destination
specific to each of them.
By applying the preceding analysis to the set of six
general equations reported at the beginning of this lesson, for the equilibrium
of a solid body animated by any forces, it is easy to
recognize that the first three relate to equilibrium of
translation, and the other three at rotational equilibrium. In the
first group, the first equation prevents translation along
the x axis, the second along the y axis, and the third along
the z axis. In the second group,
body to rotate along the x, y plane, the second along the
x, z plane , and the third along the y, z plane. We can clearly see
how the coexistence of all these equations
necessarily establishes the equilibrium.
This decomposition would still be useful to reduce, in each case,
the equilibrium equations to the strictly necessary number, when one
comes to particularize more or less the system of forces considered,
instead of supposing it to be entirely arbitrary. Without going into any
special detail here , it will suffice to say, in accordance with the
previous point of view, that, the particularization of the proposed system restricting
more or less the possible motions, either as to translation or
as to rotation, after having first determined exactly in
each case, which will always be easy, in what this
restriction consists , it will be necessary to remove, as superfluous, the
equilibrium equations relating to the translations or to the rotations which cannot
take place, and to keep only those which relate to the
motions which remain possible. It is thus that, according to the greater
or lesser limitation of the particular system of forces which one considers, it
can, instead of six equations necessary in general for equilibrium,
only three, or two, remain. , or even just one, that it will be
thereby easy to obtain in each case.
We must make perfectly analogous remarks as to the
restrictions of movement which would result, not from the
special constitution of the system of forces, but from the more or less narrow constraints to
which the body could be subjected in certain cases, and which
would produce similar effects. It would also suffice then to see
clearly which motions are made impossible by the nature of the
imposed conditions, and to suppress the equilibrium equations which
relate to them, while preserving those relating to the motions which remain free.
Thus, for example, in the case of any system of
forces, we would find that the last three equations are sufficient for
equilibrium, if the body is held by a fixed point around which it
can turn freely in any direction, any translational movement
then having become impossible; in the same way one would see the equilibrium equations to
be two in number, or even to be reduced to one, if there were at
the same time two fixed points, according to whether the body could or not slide
along the axis which seals them; and finally we would come to recognize that
equilibrium necessarily exists without any conditions, whatever
the forces of the system, if the solid body presents three
fixed points not in a straight line. Finally we could still use the same
order of considerations when the points, instead of being
rigorously fixed, would only be constrained to remain on given
curves or surfaces.
The spirit of the analysis which I have just sketched is, as we see,
entirely independent of any method according to which
the equations of equilibrium will have been obtained. But the various
general methods are far from lending themselves with the same facility to
the application of this rule. The one that best adapts to it is
undoubtedly the static method proper, founded, as
we have seen, on the principle of virtual speeds. We must
put, in fact,
principle, the perfect clarity with which he naturally analyzes the
phenomenon of equilibrium, considering distinctly each of
the elementary movements permitted by the forces of the system, and
immediately furnishing an equation of equilibrium specially relative to this
movement. The dynamic method does not have this
important advantage . It must be recognized, however, that, in the way in which M.
Poinsot conceived it, it is considerably
improved in this respect , since the only distinction between the conditions of equilibrium
relating to the forces and those relating to the couples, a
distinction which s 'then necessarily establishes, realizes by itself the
separate determination between translational equilibrium and
rotational equilibrium . But the ordinary dynamic method, exclusively used in
statics before M. Poinsot's reform, and which I characterized as a
whole at the beginning of this lesson, in no way fulfills this
essential condition, without which nevertheless it seems to me impossible
to conceive clearly the analytical expression of the general laws of
equilibrium.
Having considered the various main ways of arriving at
the exact laws of abstract equilibrium for any system of
forces, assuming the bodies in that completely passive state which
we had first recognized, though purely hypothetical,
strictly indispensable for establishing the fundamental principles
of rational mechanics; we must now examine how
geometers were able to take account of the general properties natural to
real bodies, and which must necessarily be taken into account in any
effective application of abstract mechanics. The only one that we know
so far to take into account in a really complete way,
is the gravity of the earth. Let us see how we were able to introduce it, in
fact, into the static equations. This important examination constitutes,
without doubt, in the strictly logical order of our
philosophical studies , a vicious anticipation on the part of this course
relating to physics proper, where we will
especially consider the science of gravity. But the theory of centers of
gravity, to which this static study of
terrestrial gravity is essentially reduced , plays too extensive and too important a role in
all parts of rational mechanics for us to
dispense with indicating it here, like all surveyors,
although it is not strictly regular. Moreover, I must point out
on this subject that we would almost entirely avoid everything that is
really irrational in this scientific provision, without
however depriving ourselves of the capital advantages presented by the resolution.
prerequisite of such a question, if we contracted the habit of classifying
the theory of centers of gravity among the researches of pure
geometry, as I proposed at the end of the thirteenth lesson.
To take account of terrestrial gravity, in
static questions , it suffices, as we know, to represent, in this respect,
each homogeneous body as a system of parallel and equal forces,
applied to all the molecules of the body, and of which it is necessary to
completely determine the resultant, which will from then on be introduced without any
difficulty among the primitive external forces. In reality, this
parallelism and this equality of molecular gravities are not
indeed only approximations, since, in fact, all these
forces would concur at the center of the earth if this planet were
rigorously spherical, and that their absolute intensity, independently
of the inequalities which are due to the centrifugal force produced by the
rotational motion of the earth , varies inversely with the squares
of the distances of the corresponding molecules at the center of our globe.
But, when it is only a question of the terrestrial masses at our disposal, for
which these applications of statics are ordinarily intended,
the dimensions are never large enough for the lack of
parallelism and equality between the gravities of the various molecules of
each mass, must be really taken into account. We
therefore assume , with reason, all these forces rigorously parallel and
equal, which greatly simplifies the question of their composition. In
fact, their resultant is, from that moment equal to their sum, and acts
along a line parallel to their common direction, so that its
intensity and direction are immediately known. All the
difficulty is therefore reduced to finding its point of application,
that is to say what is called the _centre of gravity_ of the body. According to
the general properties of the point of application of the resultant in
any system of parallel forces, the distance from this point to a
any plane is equal to the sum of the moments of all the forces of the
system with respect to this same plane, divided by the sum of these forces
themselves. By applying this formula to the center of gravity, and having
regard to the simplification produced by the equality of all the
proposed forces, we find that the distance from the center of gravity to
any plane is equal to the sum of the distances of all the points of
the considered body, divided by the number of these points; that is to say, that
this distance is, what is properly called the arithmetic mean
between the distances of all the proposed points. This
fundamental consideration obviously reduces the notion of the center of gravity to be
purely geometric, since by seeking it thus as _centre of
medium distances_, according to the very-rational denomination of the
ancient geometers, the question no longer retains any trace of its
mechanical origin, and consists only in this problem of
general geometry : Given a any system of points arranged between them
in a determined manner, find a point whose distance to
any plane is average between the distances of all the points given to
this same plane. There would be, as I have already indicated,
important advantages to usually conceive of the general notion of the center
of gravity in this way, completely disregarding all considerations.
gravity, because this simple and purely geometric idea is
precisely the one that must be formed in most of the
main theories of rational mechanics, especially when one considers the
large dynamic properties of the center of medium distances, where
the idea heterogeneous and superabundant gravity ordinarily introduces
a vicious complication and obscurity. This way of conceiving
the question naturally leads, it is true, to excluding it from
mechanics in order to include it in geometry, as I have
proposed. If I did not classify it thus effectively, it is only in
order to deviate as little as possible from the habits universally.
received, although I was very convinced that such a transposition
would be the only truly rational arrangement. Be that as it may with
this discussion of order, what matters essentially is not to
be mistaken as to the true nature of the question, at
whatever time and under whatever denomination one considers appropriate to
treat it.
The only geometrical definition of the center of gravity would
immediately give the means of determining it, if the system of points which
one considers was made up only of a finite number of isolated points,
because it would then directly result from very formulas. simple and which
would not need to be transformed to express the
coordinates of the point sought, relative to three rectangular axes
fixed arbitrarily. But these fundamental formulas can no longer
be used without transformation, as soon as it is a question of a system
composed of an infinity of points forming a true continuous body,
which is the ordinary case. Because the numerator and the denominator of each
formula thus becoming simultaneously infinite, these formulas no
longer offer any distinct meaning, and can only be applied
after having been suitably transformed. It is in this
general transformation that consists, from an analytical point of view, all
the fundamental difficulty of the question of the center of gravity considered.
under the broadest point of view. However, it is clear that the
integral calculus immediately gives the means to overcome it, since these
two infinite sums which constitute the two terms of each formula,
are obviously by themselves true integrals, including that which
expresses the common denominator of the three formulas relates to the
infinitely small geometric elements of the mass considered, and that
which represents the numerator proper to each formula relates to the
products of these elements by their corresponding coordinates. It follows
from there, to consider here only the most general case, that by
decomposing the body only into infinitely small elements in two
meaning by two series of infinitely close planes parallel, one
to the x, z plane, the others to the y, z plane, we will immediately find the
fundamental formulas, / [x_1 = / frac {/ iint xzdxdy} {/ iint zdxdy} , /; y_1 =
/ frac {/ iint yzdxdy} {/ iint zdxdy}, /; z_1 = / frac {1} {2} / frac {/ iint
z ^ 2dxdy} {/ iint zdydx} /] which will make known the three coordinates of the
center of gravity of the volume of a homogeneous body of any shape,
limited by a surface whose equation in x, y, and z, is assumed to be
given. We will obtain in the same way, for the center of gravity of the
surface alone of this body, the formulas / [x_1 = / frac {/ iint
xdxdy / sqrt {1 + / frac {dz ^ 2} {dx ^ 2} + / frac {dz ^ 2} {dy ^ 2}}} {/ iint
dxdy / sqrt {1 + / frac {dz ^ 2} {dx ^ 2} + / frac {dz ^ 2} {dy ^ 2}}} /]
/ [y_1 = / frac {/ iint
ydxdy / sqrt {1+ / frac {dz ^ 2} {dx ^ 2} + / frac {dz ^ 2} {dy ^ 2}}} {/ iint
dxdy / sqrt {1 + / frac {dz ^ 2} {dx ^ 2} + / frac {dz ^ 2} {dy ^ 2}}} /] / [z_1 = / frac {/ iint
zdxdy / sqrt {1 + / frac {dz ^ 2} {dx ^ 2} + / frac {dz ^ 2} {dy ^ 2}}} {/ iint
dxdy / sqrt {1 + / frac {dz ^ 2} {dx ^ 2} + / frac {dz ^ 2} {dy ^ 2}}} /] The determination of the
centers of gravity will therefore be reduced, in each particular case,
to purely analytical research, quite similar to those
required, as we have seen, for quadratures and cubatures.
However, these integrations being, in general, more complicated, the state
of extreme imperfection in which the calculation has found so far.
integral will still rarely allow a
definitive solution to be reached . But these general formulas are none the less, by
themselves, of capital importance, for introducing the consideration
of the center of gravity into the general theories of
analytical mechanics , as we shall have special occasion to
recognize soon. We must, moreover, consider, as to the question
itself, that these formulas undergo very great simplifications, when
we come to suppose that the surface which completes the proposed body is a
surface of revolution, which fortunately takes place in most the
really important applications.
This is therefore essentially the way of taking account of
terrestrial gravity in applications of abstract statics.
As to universal gravity, we can say that until now it has
been taken into consideration in a really complete way, only in
relation to spherical bodies. It is not that, when the law of
gravitation is supposed to be known, and especially by conceiving it
inversely proportional to the square of the distance, as in
true universal gravity, one cannot easily construct,
using integrals suitable, formulas which express the attraction
of a body of any figure and constitution on a given point,
and even on another body. But these general symbolic expressions
have so far remained mostly inapplicable, for want of being able to
effect the integrations which they indicate, even when we assume,
to simplify the question, that each body is homogeneous. It is
still only by a very imperfect approximation that we have been able to arrive at
the definitive determination in the very simple case of the attraction of
two ellipsoids, and the approximations could not be carried out to the
degree of precision. suitable, that by supposing these elipsoids very little
different from the sphere, which happily takes place for all our
planets. It should also be considered that, in reality, these
formulas assume prior knowledge of the law of density
within each proposed body, which we are
completely ignorant of so far .
In the present state of this important and difficult theory, we can
say that Newton's primitive theorems on the attraction of
spherical bodies indeed still constitute the most useful part of
this order of notions. These properties so remarkable, and which Newton so
simply established, consist, as we know, in that 1st
the attraction of a sphere of which all the molecules attract in
inverse ratio to the square of the distance, is the same, on a
any external point , that if the entire mass of this sphere were all
condensed at its center; 2 ° when a point is placed in the interior
of a sphere whose molecules act on it according to this same law,
it experiences absolutely no attraction from the whole portion
of the globe which is at a greater distance than him from the center, at
least, assuming, if the globe is not homogeneous, that each of its
concentric spherical layers presents the same
density at all its points .
Gravity is the only natural force that we really know how to
take into account in rational statics: we can still see how
little progress is in this study compared to universal gravity. As to
general external circumstances, from which we also had to
first completely disregard in order to establish the rational laws of
mechanics, such as friction, the resistance of the media, etc., we
can say that we do not yet know how to
introduce them in the fundamental relations given by
analytical mechanics , for we have hitherto only achieved this with the help of
very precarious hypotheses , and even obviously inaccurate, which cannot
really be considered, in the greatest number of cases, that as
suitable to provide exercises of calculation. Moreover, we will
naturally have to come back to this subject in the part of this course relating to
physics proper.
To complete the philosophical examination of the whole of statics, it
remains for us finally to consider briefly the general manner of establishing
the theory of equilibrium, when the body to which the forces are
applied is supposed to be in the state fluid, either liquid or
gas.
Hydrostatics can be completely treated according to two
perfectly distinct general methods , according to whether one seeks directly the
laws of the equilibrium of fluids according to static considerations
exclusively peculiar to this class of body, or whether one is limited to to
deduce them simply from the fundamental principles which have already provided the
static equations of solid bodies, having regard only, as
appropriate, to the new characteristic conditions which result from
fluidity.
The first method must naturally have started by being the only one
employed, as being originally the easiest, if not the most
rational. Such was indeed the character of the work of the
geometers of the seventeenth and eighteenth centuries on this important
section of general mechanics. Various static principles
peculiar to fluids, and more or less satisfactory, have been
successively proposed, mainly on the occasion of the famous
question in which geometers proposed to determine _à
priori_ the true figure of the earth, originally supposed to be any
fluid, a capital question which, considered as a whole, is
in fact linked , directly or indirectly, to all the
essential theories of hydrostatics. We know that Huyghens had first
tried to solve it, taking as the principle of equilibrium the
obviously necessary perpendicularity of gravity to the
free surface of the fluid. Newton for his part had, at the same time, chosen as a
fundamental consideration the no less obvious necessity of the equality
of weight between the two fluid columns going from the center, one to the
pole, the other to any point of l 'equator. Moving, discussing
This important question later showed clearly that these two
ways of proceeding were equally vicious, in that the principle
of Huyghens and that of Newton, although both incontestable, did not
agree, in a great number of cases, to give the same shape
to the fluid mass in equilibrium, which fully demonstrated
their common insufficiency. But Bouguer was gravely mistaken in his turn,
in believing that the union of these two principles, when they
agreed to indicate the same figure, was entirely sufficient
for equilibrium. Clairaut, in his immortal treatise _of the figure of the
earth_, was the first to discover the true general laws of
the equilibrium of a fluid mass, speaking of the obvious consideration
of the isolated equilibrium of any infinitely small channel; and, according to
this infallible _criterium_, he showed that there could exist an infinity
of cases in which the combination required by Bouguer was
observed without, however, the equilibrium taking place. Since
Clairaut's work as a whole founded rational hydrostatics,
several great geometers, continuing to adopt the same
general way of proceeding, occupied themselves with establishing the mathematical theory of
the equilibrium of fluids on more serious considerations. natural and more
distinct than that employed by its illustrious inventor. We have to
principally distinguish, in this respect, the works of Maclaurin and
especially those of Euler, who gave this fundamental theory the
simple and regular form that it now has in all
ordinary treatises , basing it on the principle of equality of pressure in
every sense, which can be regarded as a general law indicated by
observation relative to the static constitution of fluids. This
principle is undoubtedly, in fact, the most suitable that one
can employ in such a research, when one wishes to treat
directly by some consideration specific to fluids the theory of
their equilibrium, of which it immediately provides the general equations.
with extreme ease. It is then sufficient, to obtain them as
simply as possible, after having designed the fluid mass divided into
cubic molecules by three series of infinitely close planes,
parallel to the three coordinated planes, to express that each molecule
is also pressed along the three perpendicular axes. on its
faces by all the forces of the system, the pressure of the molecule
in each direction being equal to the difference of the pressures exerted on
the two corresponding opposite faces. We thus find that the
mathematical law of the equilibrium of any fluid, by whatever forces
it is requested, is expressed by the three equations:
/ [/ frac {dP} {dx} = pX, /; / frac {dP} {dy} = pY, /; / frac {dP} {dz} = pZ, /] where P
expresses the pressure supported by the molecule whose coordinates are
x, y, z, and the specific density or gravity p, and X, Y, Z, designate
the total components of the forces of which the fluid is animated along the
three coordinated axes. As we can obviously deduce, from the set
of these three equations, the formula / [P = / int p (Xdx + Ydy + Zdz) /] for
the determination of the pressure at each point, when the forces are
known as well as the law of density, it is possible to give
another analytical form to the general law of the equilibrium of fluids, by
limiting oneself to saying that the differential function, placed here under the
sign S, must satisfy the known conditions of integrability
relative to the three independent variables x, y, z, which is
precisely the very simple expression originally found by Clairaut with
regard to the mathematical theory of hydrostatics.
The study of the equilibrium of fluids constantly gives rise to a
very important new general question which is specific to them, that which
consists in determining, in the case of equilibrium, the figure of the surface
which limits the fluid mass. The abstract solution of this question is
implicitly included in the previous fundamental formula,
since it is obviously sufficient to assume that the pressure is zero or
less constant, to characterize the points of the surface, which
gives indiscriminately / [Xdx + Ydy + Zdz = 0 /] for the
general differential equation of this surface. All the concrete difficulty
is therefore essentially reduced, in each case, to knowing the real law
relating to the variation of the density in the interior of the
proposed fluid mass , unless it is homogeneous, a determination which
presents all obstacles. - quite insurmountable in
the most important applications . Leaving this aside, the question therefore only
presents a more or less complicated analytical research,
consisting in the integration, most often still unknown, of
the previous equation. It should be noted, moreover, that this equation
is, by its nature, general enough so that it can be applied even to
the equilibrium of a fluid mass which would be animated by a
determined rotational movement , as required especially the great question of the
figure of the planets. It then suffices to understand, among the
forces of the proposed system, the centrifugal forces which result from this
rotational movement.
Such is, by outline, the general manner of establishing the
mathematical theory of the equilibrium of fluids, basing it directly on
static principles peculiar to this kind of body. We design,
as I have already indicated, that this method must first have been the only one
used; for, at the time of the first researches, the
characteristic differences between solids and fluids must
necessarily appear too considerable for no surveyor to be able
to propose to apply to them the general principles
intended only for others, having only regard, in this
deduction, to some new special conditions. But, when the
fundamental laws of hydrostatics have at last been obtained, and
the human mind, ceasing to be preoccupied with the difficulty of their
establishment, has been able to measure with accuracy the real diversity that exists
between the theory of fluids and that of solids, it was impossible
that he did not seek to reduce them both to the same
essential principles , and that he did not recognize, as a general thesis, the
necessary applicability of the fundamental rules of static at
fluid equilibrium , provided that proper account is taken of the variability of
shape which characterizes them. In short, science could not remain in
this respect in its primitive state, where an
obviously exaggerated importance was attached to the conditions peculiar to fluids. But, to
subordinate hydrostatics to statics properly so called, and
thus increase by a greater unity the rational perfection of science,
it was essential that the abstract theory of equilibrium should be
treated beforehand according to a sufficiently general static principle,
which alone could be directly applied to fluids as well as to
solids, because one could not have recourse, for this purpose, to
proper equilibrium equations , in the formation of which we had
always had, necessarily, more or less regard to the invariability of the
system. This inevitable condition was fulfilled when Lagrange
conceived the way of founding statics, and consequently all
rational mechanics , on the sole principle of virtual speeds. This principle
is obviously, indeed, by its nature, just as directly
applicable to fluids than to solids, and this is one of its
most valuable properties. From then on hydrostatics, philosophically classified
in its natural rank, was no longer, in Lagrange's treatise, more than a
secondary division of statics. Although this way of
conceiving it has not yet been able to become sufficiently familiar, and the
direct hydrostatic method has so far remained the only usual one, there
is no doubt that Lagrange's method will end up being
habitually and exclusively adopted, as being that which imprints on
science its true definitive character, by making it derive
entirely from a single principle.
To represent clearly, in general, how the principle of
virtual speeds can lead to the fundamental equations of
the equilibrium of fluids, it suffices to consider that all that
such an application requires in particular consists only in understanding
among the unspecified forces of the system a new force, the pressure
exerted on each molecule, which will introduce one more term into
the general equation, or, more exactly, which will give rise to three
new virtual moments, if one distinguishes, as it should be, the
variations separately relating to each of the three coordinated axes. By
doing so, we will immediately arrive at the three equations
of fluid equilibrium, which have been reported above
from the hydrostatic method proper. If the fluid considered
is liquid, it will be necessary to design the system subject to this
characteristic condition of being able to change shape, without however ever
changing volume. This condition of incompressibility will be introduced
all the more naturally into the general equation of
virtual speeds , since it can be expressed immediately, as
Lagrange did , by an analytical formula similar to that of the terms of
this equation , by expressing that the variation of the volume is zero,
which even allowed Lagrange to represent abstractly this
incompressibility as the effect of a certain new force, of which it
suffices to add the virtual moment to those of the forces of the system. If we
want to establish, on the contrary, the theory of equilibrium for
gaseous fluids , it will be necessary to replace the condition of incompressibility by
that which subjects the volume of the fluid to vary according to a
determined function of the pressure, for example in inverse ratio to this
pressure, in accordance with the physical law on which Mariotte based
all the mechanics of gases. This new circumstance will give rise to
an equation analogous to that of liquids, although more complicated.
Only this last section of the general theory of equilibrium,
In addition to the great analytical difficulties which are specific to it, one will
necessarily feel, in applications, the uncertainty where
one is still on the true law of gases relative to the function
of the pressure which really expresses the density, because the law de
Mariotte, so precious by its extreme simplicity, can
unfortunately only be regarded as an approximation, which,
sufficiently exact for average circumstances, cannot be
rigorously extended to any case whatever.
This is the fundamental character of the undoubtedly the most
rational method which can be employed to form the abstract theory of
the balance of fluids, and which we must consider, especially in this
work, as constituting henceforth the definitive conception of
hydrostatics. This conception will appear all the more philosophical
as, in the statics thus treated, one finds a series of cases in
some way intermediate between the solids and the fluids, when one
considers the questions relating to the solid bodies likely to
change form until to a certain degree according to determined laws,
that is to say, when flexibility and
elasticity are taken into account , which establishes, in the analytical relation, a
natural filiation which leads to a succession research
almost insensitive, systems whose form is strictly
invariable to those where it is, on the contrary, eminently variable.
After having briefly examined how rational statics,
considered as a whole, could be raised at last to that high degree of
speculative perfection in which all the questions which it is likely
to present, constantly treated according to a single
directly established principle , are uniformly reduced to simple problems
of mathematical analysis, we must now undertake the same
study relatively to the last branch of general mechanics,
necessarily more extensive, more complicated, and therefore more
difficile, that which focuses on the theory of motion. This will be the
subject of the next lesson.
SEVENTEENTH LESSON.
SUMMARY. General view of the dynamics.
The essential object of dynamics consists, as we have seen, in
the study of the various motions produced by the _continuous_ forces, the
theory of the uniform motions due to the _ instantaneous_ forces being
entirely only a simple immediate consequence of the three
fundamental laws of motion. In this dynamic of varied motions or
continuous forces, we usually and with reason distinguish two
general cases , depending on whether we consider the motion of a point or that of a point.
body. From the most positive point of view, this distinction amounts to
conceiving that, in certain cases, all the parts of the body take
exactly the same movement, so that it is then in effect sufficient to
determine the movement of a single molecule, each moving as
if it were isolated, regardless of the binding conditions of the
system; while, in the most general case, each portion of the
body or each body of the system taking a distinct movement, we must
examine these various effects and know the influence exerted on them by
the relations which characterize the system considered. The second theory
being obviously more complicated than the first, it is by this one
that it is necessarily necessary to commence the special study of
dynamics, even when they are both deduced from uniform principles.
Such is also the order which we will adopt here in the indication of our
philosophical considerations.
With regard to the motion of a point, we already know that the
general question consists in determining exactly all the circumstances of
the compound curvilinear motion, resulting from the simultaneous action of
various continuous forces of any kind, supposing entirely known the
rectilinear motion which the mobile would take. under the exclusive influence
of each force considered in isolation. We also found that
this problem was susceptible, like any other, to be considered in
the opposite direction, when we proposed, on the contrary, to discover by
what forces the body is solicited, according to the
characteristic circumstances directly known to compound motion.
But, before entering into the philosophical examination of these two
general questions , we must first turn our attention to a
very important preliminary theory , that of the varied movement considered in
itself, that is to say in accordance with to the ordinary expression, the theory
of rectilinear motion produced by a single continuous force, acting
indefinitely in the same direction. This elementary theory is
indispensable to establish the fundamental notions which are reproduced
unceasingly in all the parts of the dynamics. Here is what it
essentially consists of, according to our way of conceiving
rational mechanics.
We have previously noticed that, in the
direct dynamic question , it was necessary to suppose that the effect of each
unique force was known, the real unknown of the general problem being the effect
determined by the concurrence of all the forces. This observation is
indisputable. But, according to this, what can be the object of this
preliminary part of the dynamics which one intends to the study of the movement
resulting from the action of? a single continuous force? The contradiction
apparent only depends on the imprecise expressions which are
ordinarily employed , and according to which such a question would seem as
distinct and as direct as the real dynamic questions,
while it is really only a preliminary. To
clearly conceive of its true character, it must be observed that the varied motion
produced by a single continuous force can be defined in several
ways, which depend on each other, and which, consequently, can
never be given simultaneously, although each can be
separately the most suitable, from which results the necessity of knowing how to
pass, in general, from any one of them to all the
others: it is in these transformations that the
preliminary general theory of varied motion properly consists , very
incorrectly designated under the name of the study of the action of a single force. These
various equivalent definitions of the same varied movement result from
the simultaneous consideration of the three
distinct fundamental functions , although co-relative, that can be considered there, space,
speed and force, conceived as depending on the elapsed time. The law of
motion can be immediately given by the relation between the space
traveled and the time elapsed, and then it is important to deduce from it the
_speed acquired_ by the moving body at each instant, that is to say that of the
uniform movement which would take place if, the continuous force suddenly ceasing to
act, the body no longer moved except by virtue of the
natural impulse resulting, according to the law of inertia, from the movement already
effected: it is also interesting to determine also what is, at
each instant, the intensity of the continuous force, compared to that of a
well-known constant accelerating force, such, for example, as the
terrestrial gravity, the only force of this kind which
familiar enough to us to usually serve as a suitable type. On other
occasions, on the contrary, the movement may be naturally defined
by the law which regulates the variation of the speed as a result of time, and
whence it will be necessary to conclude that relating to space, as well as that
relating to force. It would be the same if the primitive definition of
motion consisted in the law of continuous force, which might
not always be immediately given as a function of time, but
sometimes in relation to space, as for example when it s It is a matter
of universal gravitation, or at other times relatively to
speed, as can be seen from the resistance of media. Finally, if
we consider this order of questions from the most
extensive point of view , we must conceive, in general, that the definition of a
varied motion can be given by any equation, which can contain to
both these four variables of which only one is independent, time,
space, speed, and force; the problem will consist in deducing from
this equation the distinct determination of the three
characteristic laws relating to space, speed and force, as a
function of time, and, consequently, in mutual correlation. This
general problem is constantly reduced to a purely analytical research,
using the two fundamental dynamic formulas which express, as a
function of time, the speed and the force, when one supposes known the
law relating to space.
The infinitesimal method leads to these two formulas with the
greatest ease. To obtain them, it suffices to consider,
according to the spirit of this method, the motion as uniform during
the duration of the same infinitely small interval of time, and as
uniformly accelerated during two consecutive intervals. From then on, the
speed, momentarily assumed to be constant, according to the first
consideration, will naturally be expressed by the differential of
space divided by that of time; and, likewise, the continuous force,
according to the second consideration, will evidently be measured by the ratio
between the infinitely small increase in velocity, and the time
employed in producing this increase. Thus, by calling t the
elapsed time , e the space traveled, v the acquired speed and / varphi the intensity
of continuous force at each instant, the general and
necessary correlation of these four simultaneous variables will be expressed
analytically by the two fundamental formulas, / [v =
/ frac {de} {dt}, /; / varphi = / frac {dv} {dt} = /frac Danemarkd^2eacée Danemarkdt^2 rire./] According to
these formulas, all questions relating to this
preliminary theory of varied motion will immediately be reduced to simple
analytical research, which will consist either in differentiations,
or, more often, in integrations. Considering the
most general case, where the primitive definition of the proposed motion would be
given only by an equation between the four variables, the
analytical problem will consist in the integration of a
differential equation of the second order, relating to the function e, and which could
be frequently inexecutable, considering the current extreme imperfection of the
integral calculus.
Lagrange's fundamental conception, relative to
transcendent analysis , having necessarily obliged him to deprive himself of the facilities
offered by the use of the infinitesimal method for the establishment of the
two preceding dynamic formulas, he was led to present this
theory from a new point of view, the importance of which, it
seems to me, has not been sufficiently appreciated,
Lagrange showed in his _theory of analytical functions_ that this
dynamic consideration really consisted in conceiving of
any varied motion as composed at each instant of a certain
uniform motion and another uniformly varied motion, assimilating it to
the vertical motion of a heavy body launched with an initial impulse.
But, in order to give this luminous conception all its
philosophical value , I believe I should present it under a more
extended point of view than did Lagrange, as giving rise to a
complete theory of the assimilation of movements, exactly similar to the
general theory of the contacts of curves and surfaces, exposed in
the thirteenth and fourteenth lessons.
To this end, suppose any two rectilinear motions, defined
by the equations e = f (t), E = F (t); that the two mobiles have reached
the same situation at the end of time t; and consider their
mutual distance after a certain time t + h. This distance, which will be equal to the
difference of the corresponding values of the two functions f and F will
obviously have for expression, according to Taylor's formula, the series
/ [(f '(t) -F' (t)) h + (f '' (t) -F '' (t)) / frac {h ^ 2} {1.2.} + /]
/ [(f '' '(t) -F' '' (t)) / frac {h ^ 3} {1.2.3.} + / mbox {/ rm etc.} /] Using
this series, we will be able, by considerations entirely analogous to
those used in the theory of curves, to get a clear idea of
the more or less perfect assimilation of the two motions, according to the
more or less extended analytical relations of the two
primitive functions f and F. If their derivatives of the first order have the same
value, there will exist between the two movements what one could call
an _assimilation of the first order_, similar to the contact of the first
order in curves, and which we can characterize, in the
concrete relation , by saying then that the movement of the two body will be the same
for an infinitely small moment. If, moreover, the two
second-order derivatives still take the same value, the assimilation of
movements will become more intimate, and will rise to the second order; it
will then consist physically in that the two mobiles will have the same
movement during two consecutive infinitely small instans.
Similarly, by adding to these first two relations the equality of the
third derivatives, we will establish, between the movements considered, an
_assimilation of the third order_, which will make them coincide for three
consecutive _instans_, and so on indefinitely. The degree of
similarity of the two motions, determined analytically by the number of
successive derived functions which will have respectively the same value,
will always have for concrete interpretation the coincidence of the two
mobile for an equal number of consecutive instans; as we have
seen the order of the contact of the curves measured geometrically by the
community of a corresponding number of successive elements. If the law
characteristic of one of the proposed motions contains, in its
analytical expression, some arbitrary constants, we
can_assimilate it_ to any other motion up to an _order_ marked
by the number of these constants, which will then be determined by after
the equations intended to establish, according to the preceding theory, this
degree of intimacy between the two movements.
This fundamental conception leads us to perceive the possibility of
less under the abstract point of view, of acquiring a more
and more detailed knowledge of any varied movement whatever, by comparing it
successively to a series of known movements, the analytical law
of which depends on an increasing number of arbitrary constants, and
which may, consequently, have an
increasingly prolonged coincidence with it . But, just as we have seen the general theory of the
contacts of lines, applied to the measurement of the curvature one by one
, having to be effectively reduced to the comparison of
any curve whatsoever first with a straight line and then with a
circle, these two lines being the only ones that we can look at as
sufficiently known to serve usefully as a type with respect to others,
similarly the dynamic theory relating to the measurement of movements
by one another must be really limited to the
effective comparison of any varied movement, first with a uniform movement where
space is proportional to time, and then with a
uniformly varied motion where space grows due to the square of time; or
else, in order to embrace everything in a single consideration, with a
movement composed of one uniform movement, and another uniformly
varied, such as that of a heavy body animated by an initial impulse.
These two elementary movements are, in fact, as remarked
Lagrange, the only ones that we really have a notion
familiar enough with that we can apply them successfully to the measure
of everyone else. By establishing this assimilation, we find,
according to the preceding theory, that any varied movement can be at
any moment compared to that of a heavy body which would have received an
initial speed equal to the first derivative of the space traveled
considered as a function of elapsed time, and which would be animated by a
gravity measured by the second derivative of this same function, which
brings us into the two fundamental formulas obtained
above by the infinitesimal method. The proposed movement will coincide
for an infinitely small instant with the uniform motion expressed
in the first part of this comparison, and for two
consecutive instants with the uniformly accelerated motion which corresponds to the
second part. We will thus form a clear idea of the movement of the
mobile at each moment, and the way in which it varies from one moment to
another, which is strictly sufficient.
Although Lagrange's conception, as I have generalized it,
ultimately leads to the same results as the ordinary theory, it is
easy to feel its rational superiority, however, since these two
fundamental theorems, in which we have seen the term
absolute of the efforts of the human mind, relatively to the study of
various movements, can now be considered as a simple
particular application of a very general method, which allows us
abstractly to glimpse a much more perfect measurement of any
movement. varied, although powerful reasons of convenience compel us
to consider only the measure originally adopted. We can see,
from what precedes, that if nature offered us a simple
and familiar example of a rectilinear motion in which space would increase in
proportion to the cube of time, adding to our
ordinary dynamic notions the usual consideration of this movement, we
would obtain a more in-depth knowledge of the nature of
any varied movement, which could then have with the triple
movement thus composed an assimilation of the third order, which would
allow us to envisage directly, by a single sight of the mind,
the cell state for three consecutive instants, while we
are now forced to stop at two instants. From an
analytical standpoint, instead of limiting ourselves to the first two functions derived
from space relative to time, this method would amount to
simultaneously considering the third derivative, which would therefore also have
a dynamic meaning, of which it is currently devoid. In
This supposition, just as we habitually conceive the
accelerating force to represent to us the changes of speed, we
would likewise have a dynamic consideration proper to represent to us
the variations of the continuous force. Our general study of
various motions would become still more perfect if, extending this hypothesis, there also
existed a known motion in which space was
proportional to the fourth power of time, and so on. But
in reality, among the simple motions in which the space traversed is found to
increase in proportion to an entire and positive power of the time which has
elapsed, observation only lets us know the motion.
uniform produced by a single impulse and the uniformly
accelerated motion which results from the terrestrial gravity following the discovery of
Galileo, we are forced to stop at the first two degrees
of the preceding theory for the general measurement of
any varied motions . Such is the true philosophical explanation of the
method universally adopted, estimated at its real value.
I thought I should insist on this explanation, because this
fundamental conception seems to me not yet to be appreciated in a
suitable way, although it is the basis of the whole dynamic
.
After a general examination of this important preliminary theory, I
passes now to consider summarily the philosophical character of
the true direct rational dynamics, that is to say of the study of
the curvilinear motion produced by the simultaneous action of various
continuous forces whatever, continuing to suppose first that the mobile
either regarded as a point, or, which amounts to the same thing, that all the
molecules of the body taking exactly the same movement, each one
moves in isolation without being affected by its connection with the others.
We must distinguish, in general, in the curvilinear motion of a
molecule subjected to the action of any forces, two
very different cases, according to whether it is moreover entirely free, of
so as to have to describe the trajectory which will result naturally from
the combination of the proposed forces, or that, on the contrary, it is
required to move on a single curve or on a given surface.
The fundamental theory of curvilinear motion can be established as a
whole according to two very distinct modes, taking as a base
one or the other of these two cases, because each of them can be treated
directly and is at the same time susceptible to relate to
the other, the two considerations being almost equally natural depending
on the point of view where the mind takes place. Speaking of the first case, it
will suffice, to deduce the second, to look at the resistance, so much
active than passive, of the curve or the surface on which the body
is subject to remain, as a new force to be joined to those of
the proposed system, as we have seen that it is customary to do in
static. If, on the contrary, one prefers to establish first the theory of the
second case, one will then bring back the first, by considering the mobile
as forced to describe the curve which it must actually travel,
which will be entirely sufficient to form the fundamental equations, although
this curve is then originally unknown. Although this
last method is not ordinarily employed, it is appropriate, I
believe, to characterize them both here, in order to give the most
complétement possible une juste idée de la théorie générale du mouvement
curviligne, car chacune d'elles a, ce me semble, des avantages importans
qui lui sont propres. Considérons d'abord la première.
Examinant, en premier lieu, le mouvement curviligne d'une molécule
entièrement libre soumise à l'action de forces continues quelconques, on
peut former de deux manières distinctes les équations fondamentales de
ce mouvement, en les déduisant par deux modes différens de la théorie
du mouvement rectiligne. Le premier mode, qui a d'abord été le plus
employé par les géomètres, quoique, sous le rapport analytique, il ne
soit pas le plus simple, consiste à décomposer à chaque instant la
total resultant of the continuous forces which act on the moving part in
two forces, one directed along the tangent to the trajectory that it
describes, the other along the normal. Consider then, for an
infinitely small instant , the motion as rectilinear and taking place in the
direction of the tangent, according to the first fundamental law of
motion. The progression of the body in this direction will obviously be due
only to the first of these two components, to which, consequently, we
can apply the elementary formula reported above by the
rectilinear movement. This component, which is also equal to the
total accelerating force multiplied by the cosine of its inclination
on the tangent, will therefore be expressed by the second function derived from
the arc of the curve relative to time. By developing this equation
by known geometric formulas, and introducing into the calculation
the components of the total accelerating force parallel to the three
rectangular coordinate axes, we finally arrive at the three
ordinary fundamental equations of curvilinear motion The second
mode, simpler and more regular, due to Euler, and since generally
adopted, consists in immediately obtaining these equations by
directly decomposing the movement of the body at each moment, as well as the
total continuous force with which it is animated, into three others in the direction of
three coordinated axes. According to the third fundamental law of
motion, the motion along each axis being independent of the motions
along the other two is due only to the total component of the
accelerating forces parallel to this axis, so that the
curvilinear motion is thus continuously replaced by the system of
three rectilinear motions, to each of which one can immediately
apply the preliminary dynamic theory indicated above. By
naming X, Y, Z, the total components, parallel to the three axes
of x, y, and z, continuous forces which act at each
instant dt on the molecule whose coordinates are x, y, z, we obtain
thus immediately the equations / [/ frac {d ^ 2x} {dt ^ 2} =
X, /; / frac {d ^ 2y} {dt ^ 2} = Y, /; / frac {d ^ 2z} {dt ^ 2} = Z, /] which one
arrives only by a rather long computation while following the first mode.
Such are the fundamental differential equations of
curvilinear motion , according to which any questions of dynamics
relating to a body all of the molecules of which take exactly the
same motion immediately boils down to purely
analytical problems , when the data have been properly expressed. By
first considering the general direct question, which is the most
important, we propose, knowing the law of continuous forces whose
the body is animated, to determine all the circumstances of its
effective movement. For this, however this law is
given, or as a function of time, or as a function of coordinates, or as a
function of speed, it will generally suffice to integrate these three
second-order equations, which will give rise to
more or less high analytical difficulties , which the imperfection of the integral calculus
can frequently make insurmountable. The six arbitrary constants
successively introduced by this integration will be determined
, moreover, having regard to the circumstances of the initial state of the moving body, of
which the differential equations could not retain any trace. We
will thus obtain the three coordinates of the body as a function of time,
so as to be able to assign exactly its position at each moment; and
we will then find the two characteristic equations of the curve
that it describes, eliminating the time between these three expressions. As
for the speed acquired by the moving body at any time, it can therefore
also be determined from the values of its three
components, in the direction of the axes, / frac {dx} {dt}, / frac {dy} {dt},
/ frac {dz} {dt}. It is also useful to note, in this regard, that
this speed v will often be capable of being immediately calculated
by a very simple combination of the three differential equations
fundamental, which obviously gives the general formula / [v ^ 2 = 2 / int
(Xdx + Ydy + Zdz), /] with the help of which a single integration will suffice
for the direct determination of the speed, when the expression placed
under the sign / int will satisfy the known integrability conditions with
respect to the three variables x, y, z, considered as
independent. This property does not take place, no doubt, in relation to
all the possible continuous forces, nor even in relation to all
those which in fact present us with natural phenomena, since,
for example, it cannot be verified for the forces which
represent the resistance of the media, or the friction, or, in
general, as to all those whose primitive law depends on time or
on speed itself. The preceding remark is none the less
rightly regarded by geometers as having an extreme
importance for simplifying analytical research to which
the problems of dynamics are reduced, because the stated condition is
constantly verified, as it is easy to prove. , in a
very extensive particular case , which includes all the major applications of
rational dynamics to celestial mechanics, that is to say that where
all the continuous forces with which the body is animated are tendencies
towards fixed centers, acting according to any function of the
distance from the body to each center, but regardless of direction.
If, now taking in reverse the general theory of the
curvilinear motion of a free molecule, we propose to determine, on the
contrary, according to the characteristic circumstances of the
effective motion , the law of the accelerating forces which could produce it, the
This question will necessarily be much simpler from an
analytical point of view, since it will consist essentially only of
differentiations. Because it will always be possible then, by
more or less complicated preliminary research, which can only
relate to purely geometric considerations, to deduce, to
the primitive definition of the proposed movement, the values of the three
coordinates of the moving body at each instant as a function of the elapsed time; and therefore
, by differentiating these three expressions twice, we will obtain
the components of the continuous forces along the three axes, from which we
can immediately conclude the law of the total accelerating force,
of whatever nature. It is thus that we will see, in the
second section of this course, the three fundamental geometrical laws
found by Kepler for the motions of the celestial bodies which compose
our solar system, necessarily lead us to the
universal law of gravitation, which then becomes the base of all
general mechanics of the universe.
After having established the theory of the curvilinear motion of a
free molecule , it is easy to include the case where this molecule is
subject, on the contrary, to remain on a given curve. It suffices,
as I indicated, to understand then, among the continuous forces to
which the molecule is originally subjected, the total resistance
exerted by the proposed curve, which will obviously allow to
consider the mobile as entirely free. All the difficulty specific
to this second case therefore essentially boils down to analyzing
this resistance with accuracy. Now, for this purpose, we must
first distinguish , in the resistance of the curve, two very different parts.
that we could call, to characterize them clearly, one
_static_, the other _dynamic_. The _static_ resistance is that which
would take place even when the body is still; it comes from the
pressure exerted on the curve proposed by the accelerating forces with
which it is driven; thus one will obtain it by determining the component of
the total continuous force according to the normal to the curve given at the point
which one considers. The _dynamic_ resistance has an entirely
different origin ; it is generated only by movement, and results from the
perpetual tendency of the body to abandon the curve which it is forced to
describe, in order to continue to follow, by virtue of the first law
fundamental of movement, the direction of the tangent. This second
resistance, which manifests itself in the passage of the body from one element of the
curve to the next element, is obviously directed at each moment
according to the normal to the curve located in the osculating plane, and may,
consequently, n 'not have the same direction as the
static resistance , if the osculating plane does not contain the straight line along
which the total accelerating force acts. It is to this
dynamic resistance that we generally give the name of _centrifugal force_, due
to the fact that the only accelerating forces considered first by
geometers were _centripetal_ forces, or tendencies towards
fixed centers. As for its intensity, by conceiving this
centrifugal force as a new accelerating force, it will be measured by
the normal component produced, in each infinitely small instant,
the speed of the moving body, when it passes from one element of the curve to one.
other. One thus easily finds, after having eliminated the
auxiliary infinitesimals initially introduced naturally by this consideration,
that the centrifugal force is continuously equal to the square of the
effective speed of the moving part divided by the corresponding radius of curvature of the
proposed curve. Moreover, this fundamental expression, as well as
the direction of the centrifugal force itself, could be entirely
obtained by calculation, by first introducing this force, in a
completely indeterminate manner, into the three
general differential equations of curvilinear motion reported above.
However that may be, after having determined the dynamic resistance, we will
compose it suitably with the static resistance, and, by
including the total resistance among the proposed forces, the problem will be
immediately brought back to the preceding case. The most remarkable question
of this kind consists in the study of the oscillatory motion of a body
weighing on any curve (and particularly on a circle or
on a cycloid), the philosophical examination of which must naturally be
referred to the part of this course dealing with physics proper
.
It would be superfluous here to consider separately the case where the mobile,
instead of having to describe a given curve, would only be required to
remain on a certain surface. It is essentially by the same
considerations that this new case, which is moreover not very important
in applications, is reduced to that of a free body. There is no other
real difference than in that then the trajectory of the mobile is not
first entirely determined, and that one is obliged, in order to know it,
to add to the equation of the surface proposed another equation
provided by the dynamic study of the problem.
Let us now consider, by outline, the second general mode
previously distinguished for constructing the fundamental theory of the
curvilinear motion of an isolated molecule, starting, on the contrary, from the case where the
molecule is previously subject to describing a given curve.
The real difficulty then consists in directly establishing the
fundamental theorem relating to the measurement of the centrifugal form. Now,
this can be done easily, by first considering the
uniform movement of the body in a circle, by virtue of an initial impulse, and
without any accelerating force, as Huyghens has supposed, to which
is due the basis of this theory. The centrifugal force is therefore
evidently proportional to the sine-verse of the arc of a circle described in
an infinitely small instant, suitably compared to the
corresponding time , from which it is easy to conclude, as Huyghens did,
that it has for expression the square of the constant speed with which
the mobile describes the circle divided by the radius of this circle. This result
once obtained, by combining it with another fundamental notion due
to Huyghens, we immediately deduce the value of the centrifugal force
in any curve. It suffices, for that, to conceive that the
determination of this force requiring only the
simultaneous consideration of two consecutive elements of the proposed curve, the
movement can be considered continuously as taking place in the
corresponding osculating circle, since this circle presents relatively
to the curve two successive common elements. We can therefore directly
transport to any curve the expression of the centrifugal force
originally found for the case of the circle, and establish, as in the
first method, but much more simply, that it is generally
equal to the square of the divided speed by the radius of the osculating circle.
This way of proceeding has the advantage of giving a clearer idea
of the centrifugal force.
The case of movement in a determined curve being thus treated
beforehand with all the suitable generality, it is easy to
bring back to it that of an entirely free body, describing the trajectory which
must naturally result from the simultaneous action of certain
accelerating forces of any kind. It suffices, in fact, according to the
previously expressed indication , to conceive of the body as subject to remaining on
the curve that it will actually describe, which obviously amounts to the same thing,
since it matters little, in dynamics, the body not being able to point
really to traverse any other curve, that it is constrained there by
the nature of the forces with which it is animated, or by special conditions of connection
. Therefore this movement will give birth to a real force
centrifugal, expressed by the general formula found above.
Now it is clear that, if the total continuous force with which the moving part
is animated was first conceived as broken down at each moment into two
others, one directed along the tangent to the trajectory, and the other
according to the normal located in the osculating plane, the latter must
necessarily be equal and directly opposed to the centrifugal force.
Now, this normal component having for expression the
total continuous force multiplied by the cosine of the angle that its direction forms with
the normal, by equaling this value to that of the centrifugal force, we
will form a fundamental equation from which we can deduce the equations
general data of curvilinear motion previously obtained by another
method. For this, we will not have to do any other transformation than
to introduce into this equation, instead of the total continuous force and
its direction, its components according to the three coordinated axes, and to
replace, in the formula which expresses the centrifugal force, the speed
and the radius of curvature by their general values as a function of the
coordinates. The equation thus obtained will naturally break down into
three, if we consider that, having to take place for any system
of accelerating forces and for any trajectory,
it must be verified separately with respect to each of the three
coordinates, considered momentarily as three entirely
independent variables . These three equations will be found to be exactly the
same as those reported above. Although this way of
obtaining them is much less direct, and it requires a greater
analytical apparatus , I nevertheless thought it necessary to indicate it distinctly,
because it seems to me suitable for shedding light, in a very
important respect , the ordinary theory of curvilinear motion, by making
sensitive the existence of centrifugal force, even in the case of a
free body , a notion on which the method usually adopted today
commonly leaves much uncertainty and obscurity.
Having sufficiently studied, in what precedes, the general character of
the part of the dynamics relating to the motion of a point, or, what
amounts to the same thing , of a body of which all the molecules move
identically, we must now examine , under a similar point
of view, the part of the dynamics most difficult and the most extended,
that which relates to the more real case of the movement of a system of
bodies linked to each other in some way, and whose movements
proper are altered by the conditions dependent on their binding. I will
consider carefully, in the following lesson, the
general results obtained so far by the surveyors, relative to this order
research. I must therefore limit myself strictly here to characterizing the
general method by which we have succeeded in converting all
problems of this nature into pure questions of analysis.
In this last part of the dynamics, it is necessary beforehand to
establish a new elementary notion, relative to the measurement of
forces. Indeed, the forces considered until now being always
applied to a single molecule, or at least all acting on the
same body, their intensity was found to be sufficiently measured,
having only regard to the speed more or less great than they
could. print to mobile at any time. But, when it comes to
To consider simultaneously the motions of several different bodies,
this way of measuring the forces obviously becomes insufficient,
since one cannot dispense with taking into account the mass of each
mobile, as well as its speed. To take it properly into
consideration, geometers have established this fundamental notion that
the forces capable of imparting the same speed to various masses
are exactly like these masses to each other; or, in other words,
that the forces are proportional to the masses, as well as we
recognized them, in the fifteenth lesson, according to the third
physical law of motion, to be proportional to the speeds. All
phenomena relating to the communication of motion by shock, or in
any other way, have constantly confirmed the supposition of this
new proportionality. It obviously follows that when it is necessary to
compare, in the most general case, the forces which impart
different speeds to unequal masses, each of them must be
measured according to the product of the mass on which it acts by the
corresponding speed. This product, to which geometers have
commonly given the name of _quantity of movement_, determines exactly, in
fact, the force of impulse of a body in the shock, the _percussion_
proper, as well as the _pressure_ that a body can exert. against
any fixed obstacle to its movement. Such is the new
elementary notion relative to the general measurement of forces, of which it would
perhaps be convenient to make a fourth and last fundamental law
of motion, at least insofar as this notion is not really
susceptible, like some geometers. 'have thought, to be logically
deduced from the preceding notions, and can be firmly established
only on physical considerations which are proper to it.
This preliminary notion being established, let us now examine the
general conception according to which the dynamics of
any system of bodies subjected to the action of any forces can be treated . The
The characteristic difficulty of this order of questions consists
essentially in the manner of taking into account the connection of the
different bodies of the system, by virtue of which their
mutual reactions will necessarily alter the proper movements which each
body would take, if it were alone, by the influence of the forces which
solicit it, without one knowing at all _a priori_ of what
this alteration may consist of. Thus, to choose a very simple,
and nevertheless important, example in the famous problem of the motion of a
compound pendulum, which was originally the principal subject of the
researches of geometers on this higher part of dynamics, it is
is evident that, owing to the connection established between the bodies or
molecules nearest to the point of suspension, and the bodies or
molecules which are farthest from it, there will be a reaction
such that neither some nor others will not oscillate as if they were
free, the movement of the former being delayed, and that of the latter
being accelerated by virtue of the necessity in which they find themselves to oscillate
simultaneously, without any dynamic principle already established being able to make
known the law which determines these reactions. It is the same in
all other cases relating to the movement of a system of bodies. We
therefore obviously feel here the need for new designs
dynamic. The geometers, obeying this subject, the habit imposed
almost constantly by the weakness of the human mind, first
dealt with this new series of researches, creating, so to speak, a
new particular principle in relation to each
essential question . Such were the origin and the destination of the various
general properties of motion which we will examine in the
following lesson , and which, originally considered as so many _principles_
independent of each other, are no longer today, in the eyes of
geometers, as remarkable theorems provided simultaneously by the
fundamental dynamic equations. We can follow, in the _Mechanics
analytique_, the general history of this series of works, which Lagrange
has presented in a way so deeply interesting for the study of
the progressive march of the human mind. This way of proceeding
was continually adopted until d'Alembert, who put an end to all
this isolated research, by amounting to a general conception of
how to take account of the dynamic reaction of the bodies of a system
in by virtue of their connections, and consequently establishing the
fundamental equations of motion of any system. This conception,
which has always been used since, and which will serve indefinitely as a basis for
all research relating to the dynamics of bodies, consists
essentially to include questions of movement in
simple questions of equilibrium, with the help of this famous general principle to
which the unanimous agreement of geometers has given, with so much reason, the
name of d'Alembert's principle. So let us now consider this principle
in a straightforward manner.
When, by the reactions which various bodies exert on each
other by virtue of their connection, each of them takes a movement
different from that which the forces by which it is animated would have imparted to
it had it been free, we can obviously to regard natural movement
as broken down into two, one of which is that which will actually take
place, and the other of which, consequently, has been destroyed.
d'Alembert properly consists in that all the motions of this
last kind, or, in other words, the quantities of motions
lost or gained by the different bodies of the system in their
reaction, are necessarily made equilibrium, having regard to the
binding conditions which characterize the proposed system. This
luminous general conception was first glimpsed by Jacques
Bernouilli in a particular case; for such is evidently the
consideration which he employs in solving the problem of the
compound pendulum , when he considers the quantity of movement lost by the body
nearest to the point of suspension, and the quantity of movement
gained by the one who is furthest from it, as having necessarily to
satisfy the law of equilibrium of the lever, relative to the point of
suspension, which leads him to immediately form an equation
capable of determining the center of oscillation of the system of weights the
simplest. But this idea was, for Jacques Bernouilli, only an
isolated artifice which does not detract from the merit of the great conception of
d'Alembert, whose essential property consists in its entire
necessary generality.
By considering d'Alembert's principle from the most
philosophical point of view , we can, it seems to me,
fifteenth lesson), established by Newton under the name of equality of
reaction to action. D'Alembert's principle coincides exactly, in
fact, with this law of Newton, when one considers only a system
of two bodies, acting on each other along the line which joins them.
This principle can therefore be considered as the greatest
possible generalization of the law of reaction equal and contrary to action; and
this new way of conceiving it seems to me suitable for bringing
out its true nature, thus giving it a
physical character , instead of the purely logical character which had been
imprinted on it by d'Alembert. As a result we will not see in this
great principle that our second law of motion extended to
any number of bodies, arranged between them in any way.
According to this general principle, it is conceived that any question of dynamics
can be immediately converted into a simple question of statics,
since it will suffice to form, in each case, the equilibrium equations
between the destroyed motions; which gives the necessary certainty to be
able to put into equation any problem of dynamics, and to
make it thus depend only on analytical research. But the form
in which d'Alembert's principle was originally conceived is
not the most suitable for easily carrying out this
fundamental transformation, given the great difficulty that we
often experience in discerning which movements must be destroyed, as we
can fully convince ourselves of by the careful examination
of d'Alembert's _Treaty of dynamics_, the solutions of which are usually so
complicated . Hermann, and above all Euler, sought to eliminate
the embarrassing consideration of quantities of motion lost or
gained, by replacing the motions destroyed by the
primitive motions composed with the effective motions taken in the opposite direction,
which obviously amounts to the same thing, since, when a force has been
broken down into two, we can reciprocally substitute for one of the
components the combination of the resultant with the other component
taken in the opposite direction. From then on d'Alembert's principle, considered
from this new point of view, consists simply in that the
effective motions conforming to the connection of the bodies of the system will
necessarily, being taken in the opposite direction, always strike a balance with
the primitive motions which would result from the sole action of the forces
proposed on each supposedly free body; which can moreover be
established directly, for it is evident that the system would be in
equilibrium if we imprinted on each body a quantity of movement equal
and contrary to that which it will actually take. This new shape
given by Euler to d'Alembert's principle is the most suitable for
making use of it, as taking into consideration only the
primitive motions and the effective motions, which are the real elements of the
dynamic problem, some of which constitute the data and others.
the unknowns. This is, in fact, the definitive point of view from which
d'Alembert's principle has been usually conceived since.
As questions relating to movement are thus generally reduced,
in the simplest possible way, to pure questions of equilibrium,
the most philosophical method of dealing with rational dynamics
consists in combining d'Alembert's principle with the principle of
virtual speeds, which directly provides, as we saw in
the previous lesson, all the equations necessary for the equilibrium
of any system. Such is the combination conceived by Lagrange,
and so admirably developed in his _Mécanique analytique_, which
raised the general science of abstract mechanics to the highest degree
of perfection that the human mind can aspire to in
logical terms, that is to say - to say to a rigorous unity, all the questions which
can relate to it being from now on uniformly attached to a
single principle, according to which the final solution of
any problem whatever presents only difficulties any more
analytical. To establish as simply as possible the
general formula of dynamics, let us conceive that all the accelerating forces
of the unspecified system proposed have been decomposed parallel to the
three axes of the coordinates, and are X, Y, Z, the groups of forces
corresponding to the axes of x , y, z; by designating the mass of the
system by m , there must be equilibrium, according to d'Alembert's principle,
between the primitive quantities of motion mX, mY, mZ, and the
effective quantities of motion taken in the opposite direction, which will be obviously
expressed by -m {d ^ 2x} / over {dt ^ 2}, -m {d ^ 2y} / over {dt ^ 2},
-m {d ^ 2z} / over {dt ^ 2}, according to the three axes. So, applying to this
set of forces the general principle of virtual speeds, taking
care to distinguish the variations relating to the different axes, we
will obtain the equation / [/ int m / left (X- / frac {d ^ 2x} {dt ^ 2} / right) / delta x +
/ int m / left (Y- / frac {d ^ 2y} {dt ^ 2} / right) / delta y + /] / [/ int
m / left (Z- / frac {d ^ 2z} {dt ^ 2} / right) / delta z = 0, /] which can be regarded
as implicitly comprising all the equations necessary for
the entire determination of the various circumstances relating to the
motion of any system of bodies stressed by
any forces . The explicit equations will be deduced suitably, in
each case, from this general formula, by reducing all the
variations to the smallest possible number, according to the
binding conditions which will characterize the proposed system, which will provide as many
distinct equations as there will remain truly
independent variations .
In order to bring out, from a philosophical point of view, all the
fruitfulness of this formula, and to show that it
rigorously understands the totality of the dynamics, it should be
noted that one could even draw from it, like a simple special case,
the theory of the curvilinear motion of a single molecule; which we
specially considered in the first part of this lesson. Indeed
it is obvious that, if all the proposed continuous forces act
on a single molecule, the mass m disappears from the
previous general equation , which, by distinguishing separately the virtual motion relative
to each axis, immediately provides the three fundamental equations
established above for the motion of a point. But, although we must
consider this filiation, without which we would not conceive of the whole
real extent of the general formula of dynamics, the theory of
motion of a single molecule does not really require the use of the
principle of d'Alembert, which is primarily intended for the
dynamic study of body systems. This first theory is too simple
in itself, and results too immediately from the fundamental laws of
movement, so that I did not believe it necessary, in accordance with
ordinary usage , to present it first in isolation, in order to make clearer
the important general notions to which it gives rise,
although we must end by bringing it in, in view of a
more perfect coordination, in the invariable formula which
necessarily contains all the possible dynamic theories.
It would be departing from the natural limits of this course to indicate here
no special application of this general formula to the
effective solution of any dynamic problem, the method having to be the
sole essential object of our philosophical considerations, except
the indication of the principal results which it produced, and which
we will deal with in the next lesson. However, I believe I should
recall on this subject, as a conception really relating to
_method_ much more than to _science_, the necessary distinction,
pointed out in the preceding lesson, between the movements of _translation_
and the movements of _rotation_. To properly study the motion
of any system, it is necessary, in fact, to consider it as composed
of a translation common to all its parts, and of a rotation specific
to each of its points around a certain constant axis. or variable. For
reasons of analytical simplification which we will have occasion, in
The following lesson, to indicate the origin, the geometers
always consider preferably the rotational movement of any system
relative to its center of gravity, or, to say better, to its center
of the mean distances, which presents, under this relation,
very remarkable general properties , the discovery of which is due to Euler. Consequently,
the complete analysis of the motion of a system animated by
any forces whatever essentially consists: 1 ° in determining at each moment
the speed of the center of gravity and the direction in which it is
moving, which is sufficient to make known, as we we will see
everything that concerns the translation of the system; 2 ° to be determined
also at each instant the direction of the instantaneous axis of rotation
passing through the center of gravity, and the speed of rotation of each
part of the system around this axis. It is clear, in fact, that all
the secondary circumstances of the movement can necessarily be
deduced, in each case, from these two main determinations.
The general formula of dynamics, established above, is evidently,
by its nature, just as directly applicable to the movement of
fluids as to that of solids, provided that proper
consideration is given to the conditions which characterize the fluid state. , either
liquid or gaseous, which we have had occasion to indicate in the
previous lesson about balance. Also d'Alembert, after having
discovered the fundamental principle which allowed him, given the progress of
statics, to treat as a whole the dynamics of
any system , he immediately applied it to the establishment
of general equations of motion of fluids, entirely unknown
until then. These equations are obtained especially with great ease
from the principle of virtual speeds, as expressed by
the preceding general formula. This part of the dynamic
therefore really leaves nothing to be desired in the concrete respect, and presents
only purely analytical difficulties, relating to integration.
partial difference equations which we arrive at. But it
must be recognized that this general integration hitherto offering
insurmountable obstacles, the actual knowledge that can be
deduced from this theory is still extremely imperfect, even in
the simplest cases; which will undoubtedly seem inevitable,
considering the great complication that we have already recognized in this
regard in questions of pure statics, the nature of which is however
much less complex. The only problem of the flow of a
heavy liquid through a given orifice, however easy it may seem, has not
yet been solved in a really satisfactory manner. In order to
To sufficiently simplify the analytical research on which it depends, the
surveyors were obliged to adopt the famous hypothesis proposed by
Daniel Bernouilli under the name of _parallelism of the slices_, which allows
to consider the movement only by slices, instead of having
to consider it molecule to molecule. But this assumption, which consists in
looking at each horizontal section of the liquid as moving in its
entirety and taking the place of the next, is obviously in
formal contradiction with reality in almost all cases,
except in a small number of circumstances so chosen.
expressly say , because of the lateral movements of which such a hypothesis
completely abstract, and whose sensible existence
necessarily imposes the law of studying in isolation the motion of each
molecule. The general science of hydrodynamics can therefore only
really be considered as being at its birth, even in
relation to liquids, and all the more so in relation to gases.
But it is eminently important to recognize, on the other hand, that all the
great works which remain to be done in this respect consist
essentially in the progress of mathematical analysis alone, the
fundamental equations of the motion of fluids being irrevocably
established.
After having considered under its various principal aspects the character
general of the method in rational mechanics, and indicated how
all the questions which it can offer reduce to
purely analytical researches , it remains for us now, to complete the
philosophical examination of this fundamental science, to consider, in the
following lesson , the main results obtained by the human mind by
doing so, that is to say the most
remarkable general properties of balance and movement.
EIGHTEENTH LESSON.
SUMMARY. Considerations on general theorems of
rational mechanics .
The purpose and spirit of this work, as well as its natural scope,
necessarily preclude us here from any special development relating to
the application of the fundamental equations of balance and
motion, to the effective solution of any
particular mechanical problem . Nevertheless, one would only form an incomplete idea of
the philosophical character of rational mechanics considered as a
whole, if, after having properly studied the method, one did not
finally consider the great theoretical results of science,
that is - tell the main general properties of equilibrium and
motion discovered hitherto by geometers, and which
now remain to be examined. These various properties were originally conceived
as so many true _principles_, each of which was
originally intended to provide the solution of a certain order of
new mechanical problems, superior to the methods known
until then. But, since the whole of rational mechanics has
taken on its definitive systematic character, each of these old
_principles_ has been reduced to being no more than a simple more or
less general _theorem_ , a necessary result of the fundamental theories of
statics and of abstract dynamics: it is only under this
philosophical point of view that we must consider them here. Let's start with those
which relate to statics.
The most remarkable theorem which has been deduced so far from the
general equations of
originally discovered by Torricelli, relative to the balance of
heavy bodies. It consists properly in that, when
any system of heavy bodies is in its situation of equilibrium, its center
of gravity is necessarily placed at the lowest or highest
possible point, compared to all the positions it could take.
according to any other situation in the system. Torricelli first presented
this property as immediately verified by the
known equilibrium conditions of all the weight systems considered
until then. But the general considerations from which he
then attempted to demonstrate it directly are really little.
satisfactory, and offer a sensible example of the need to be
wary, in the mathematical sciences, of any idea whose character
is not perfectly precise, however plausible it may
appear. In fact, Torricelli's reasoning
essentially consists in noting that the natural tendency of the weight being to
descend, there will necessarily be equilibrium if the center of gravity is
placed as low as possible. The insufficiency of this consideration
is obvious, since it does not explain why there is also
equilibrium when the center of gravity is placed as high as possible, and
it would even tend to demonstrate that this second case of equilibrium does not exist. can
to exist, while, from the theoretical point of view, it is as real
as the first, although, by want of stability, we seldom have
occasion to observe it in practice. Thus, to choose a
very simple example , the law of equilibrium of a pendulum requires that the center of
gravity of the weight be placed on the vertical led by the point of
suspension, which offers a palpable verification of
Torricelli's theorem ; but, when we disregard stability, it is
evident that this center of gravity can moreover be indifferently
above or below the point of suspension, equilibrium
also taking place in both cases.
The true general proof of Torricelli's theorem consists in
deducing it from the fundamental principle of virtual speeds, which
immediately provides it with the greatest ease. It suffices, in fact,
for this, to apply this principle directly to the equilibrium of
any system of heavy bodies, with respect to which it immediately gives
the equation / [/ int Pdz = 0, /] where P denotes any of the weights, and z
the vertical height of its center of gravity. Now, according to the
general definition of the center of gravity of any system of weights, we obviously have
by naming P. the total weight of the system, and z, the vertical ordinate of
its center of gravity, the relation / [/ int Pdz = P_1dz_1./] So
the equation of virtual speeds becomes, in this case, dz_1 = 0; it
can, according to the general theory of analytic and _maxima_
_minima_ immediately shows that the vertical height of the center
of gravity system is then a _maximum_ or _minimum_, as
indicated by theorem Torricelli.
This important property, independently of the great interest which it
presents from the physical point of view, can even be advantageously
employed to facilitate the general solution of several
essential problems of rational statics, relative to heavy bodies.
Thus, for example, it suffices for the entire resolution of the famous
question of the _chaînette_, that is to say of the figure assumed by a
heavy chain suspended from two fixed points, and then freely
abandoned to the sole influence of gravity, supposing it to be
perfectly flexible, and moreover inextensible. In fact,
Torricelli's theorem indicating that the center of gravity must be placed as
low as possible, the problem immediately belongs to the
general theory of isoperimeters, indicated in the eighth lesson, since it
is reduced to determining, among all the curves of the same contour drawn
between the two given fixed points, which one enjoys this
characteristic property, that the vertical height of its center of
total gravity is a _minimum_, condition which suffices to determine
completely, by means of the calculation of the variations, the
differential equation , and then the finite equation of the sought curve. The
same is true in a few other interesting questions relating to
weight balance.
Torricelli's theorem later
underwent an important generalization by the work of Maupertuis, who, under the name of _loi
du repos_, discovered a very extensive property of equilibrium, of which
that above considered is no more than 'just a special case.
It is only to terrestrial gravity, or to gravity itself
, that the law found by Torricelli applies.
extends, on the contrary, to all the attractive forces which can make
the bodies of any system tend towards fixed centers, or
towards each other, according to any function of distance,
independent of direction, which includes all the great
natural forces . We know that, in this case, the expression P / delta p + P '/ delta
p' + etc., which forms the first member of the general equation of
virtual velocities, necessarily always happens to be an
exact differential. Consequently, the principle of
virtual speeds then properly consists in that the variation of its
integral is zero, which obviously indicates, according to the theory
fundamental of the _maxima_ and _minima_, that this integral / int P / delta
p is constantly, in the case of equilibrium, a _maximum_ or a
_minimum_. This is what Maupertuis' law consists of, considered
from the most general point of view, and thus deduced directly with
extreme simplicity from the fundamental principle of virtual speeds,
which must necessarily include implicitly all the properties to
which the balance theory.
Maupertuis' theorem was presented by Lagrange under a more concrete and
more remarkable aspect , by relating it to the notion of _forces vives_, which
we will deal with below. Lagrange, considering that the integral
/ int P / delta p considered by Maupertuis is necessarily always,
according to the general analytic theory of motion, the complement of the
sum of the living forces of the system to a certain constant, concluded
that this sum of living forces is a _minimum_ when the
preceding integral is a _maximum_, and vice versa. According to this,
Maupertuis' theorem can be considered more simply as
consisting in that the equilibrium situation of any system
is constantly one in which the sum of the living forces happens to
be a _maximum_ or a _minimum_. It is evident that, in the
particular case of terrestrial gravity, this law coincides exactly
with that of Torricelli, the living force then being equal, as we
know, to the product of the weight by the vertical height of the center of
gravity, which must therefore necessarily become a _maximum_ or a
_minimum_, if there is equilibrium.
Another very remarkable general property of equilibrium, which may
be regarded as the indispensable complement of the theorem of
Torricelli and Maupertuis, consists in the fundamental distinction
between cases of _stability_ or_instability_ of equilibrium. We know that
the equilibrium can be _stable_ or _instable_, that is to say that the body,
infinitely little removed from its equilibrium situation, can tend to y
to return, and indeed returns to it after a certain number of oscillations
soon annihilated by the resistance of the environment, friction, etc., or
else it tends, on the contrary, to move away from it more and more, so as not to
stop only in a new stable equilibrium position. What we
physically call the state of _repos_ of a body is really
nothing other than_stable equilibrium_, for the abstract _repos_, as
geometers conceive it, when they suppose a body which would not be
solicited by any force, obviously could not exist in
nature, where there can only be more or less lasting balances.
The _unstable_ equilibrium, on the contrary,
vulgar properly calls _equilibre_, which always designates a
more or less transient and artificial state. The general property which we
now consider, and whose complete proof is due to
Lagrange, consists in the fact that, in any system, the equilibrium
is _stable_ or _instable_, depending on whether the integral considered by
Maupertuis, and which has been indicated , above, happens to be a
_minimum_ or a _maximum_; or, which amounts to the same thing, as we
have said, depending on whether the sum of the living forces is a _maximum_ or
a _minimum_. This beautiful theorem of mechanics, applied to the
simplest and most remarkable case, to that of the equilibrium of heavy bodies.
considered by Torricelli, then learns that the system is in a state
of stable equilibrium, when the center of gravity is placed as low as
possible, and in a state of unstable equilibrium when, on the contrary, the
center of gravity is placed as high as possible, which is easy to
check directly for the less complicated systems. Thus, for
example, the equilibrium of a pendulum is obviously stable when the center
of gravity of the weight is located above the point of
suspension, and unstable when it is below. Likewise, an ellipsoid
of revolution, placed on a horizontal plane, is in stable equilibrium
when it rests on the top of its minor axis, and in unstable equilibrium
when it is on the top of its major axis. Observation alone would have
sufficed to distinguish the two states in such
simple cases . But the deepest theory was needed to reveal
to geometers that this fundamental distinction was also
applicable to the most compound systems, by showing that when
the integral relative to the sum of the virtual moments is a _minimum_, the
system cannot go around of its situation of equilibrium only
very small oscillations, the extent of which is determined, while,
if this integral is, on the contrary, a _maximum_, these oscillations
can acquire and in fact do acquire a finite and unspecified extent.
It is, moreover, useless to warn that, by their nature, these
properties, as well as the preceding ones, take place in fluids just
as well as in solids, which is also the character of
all the general mechanical properties. examination of which we
have intended this lesson.
Let us now consider the general theorems of mechanics relating to
motion.
Since these properties have ceased to be considered as so many
_principles_, and since we have seen only the simple necessary results of
fundamental dynamic theories, the most direct and
suitable way of establishing them is to present them , as did
Lagrange, as immediate consequences of the general equation of
dynamics, deduced from the combination of d'Alembert's
principle with the principle of virtual speeds, as we exposed it in the
previous lesson. One must put among the most
significant advantages of this method, as Lagrange rightly remarked, this
facility which it offers for the demonstration of these great theorems of
dynamics in their greatest generality, a demonstration to which one
could not otherwise achieve only by indirect and
very complicated considerations. Nevertheless, the nature of this course prevents us
from specifically indicating here each of these demonstrations, and we
we must limit ourselves to considering only the various results.
The first general theorem of dynamics is that which Newton
discovered in relation to the motion of the center of gravity of
any system , and which is usually known as the _principle of
the conservation of motion of the center of gravity_. Newton was the
first to recognize and demonstrated by extremely simple considerations, at the
beginning of his great treatise on the _mathematical principles of
natural philosophy_, that the mutual action of the bodies of a system on
one another, either by attraction or by impulse, in a word
in any way, with proper regard for equality
constant and necessary between reaction and action, can in no way
alter the state of the center of gravity, so that, if there are no
accelerating forces other than these reciprocal actions, and if the
external forces of the system are reduced only to
instantaneous forces , the center of gravity will always remain stationary or will
move uniformly in a straight line. D'Alembert has since generalized
this property, and proved that, whatever alteration may be introduced by
the mutual action of the bodies of the system in the motion of each
of them, the center of gravity is never affected, and that its
movement constantly takes place as if all the forces of the system there
were directly applied parallel to their direction, regardless of
the external forces of this system, and assuming
only that it has no fixed point. This is easily
demonstrated by developing, in the general formula of dynamics, the
equations relating to the translational motion, which, by the
fundamental analytical property of the center of gravity, are found to coincide with
those which would have provided the isolated motion of this center if the
total mass of the system had been supposed to have been condensed there, and if it had been conceived to be
animated by all the external forces of the system. The main
advantage of this beautiful theorem is that it can thus, as regards the
movement of the center of gravity, to bring the case of a body or
any system into that of a single molecule. As the
translational movement of a system must be estimated by the movement of its
center of gravity, we thus succeed in this way to reduce the
second part of the dynamics to the first for all that
relates to the translational movements, d 'which results, as it is
easy to feel, an important simplification in the solution of
any particular dynamic problem, since one can then neglect, in
this part of the research, the effects of the mutual action of all
the proposed bodies, the determination of which usually constitutes the
main difficulty of each question.
We do not commonly get a sufficiently correct idea of the entire
theoretical generality of the great results of rational mechanics,
which are necessarily applicable, by themselves, to all orders of
natural phenomena, since we have recognized that the
fundamental laws on which rests the whole systematic edifice of
science, do not suffer exception in any class of
phenomena, and constitute the most general facts of the
real universe , although one usually seems, in this kind of conceptions, to
have only in view the inorganic world. Also is it apropos, this
seems to me, to point out formally here, with regard to this
first general property of motion, that the theorem also takes
place in living bodies as in inanimate bodies. Whatever
may be, in fact, the nature of the phenomena which characterize
living bodies, they can only consist at most in certain
particular actions of molecules on one another, which would not be
observed in gross bodies, without one having to doubt
, moreover, that the reaction is always there, as well as in any other
case, on the contrary equal to the action. Thus, by the very nature of the
theorem we have just considered, it must necessarily be
check for living bodies as well as for gross bodies,
since the movement of the center of gravity is independent of these
mutual interior actions . It follows, for example, that a living body,
whatever the internal play of its organs, cannot of itself
displace its center of gravity, although it can cause
certain movements to execute at some of its points. partial around this
center. Do we not clearly verify that the
total locomotion of a living body would be entirely impossible without the
external help provided by the resistance and the friction of the ground on
which it moves, or of the fluid which contains it? ? We can do
remarks which are exactly analogous, with regard to all the other
general dynamic properties which remain to be considered, and for
each of which I will therefore dispense with indicating
especially its necessary applicability to living bodies as well
as to inert bodies.
The second general theorem of dynamics consists in the famous and
important _principle of areas_, the first idea of which is due to Kepler,
who very simply discovered and demonstrated this property for the case of the
motion of a single molecule, or in other words , of a body whose
all points move identically. Kepler establishes, by the
most elementary considerations, that if the accelerating force
total of which a molecule is animated constantly tends towards a fixed point,
the vector radius of the mobile describes around this point equal areas in
equal times, so that the area described at the end of
any time increases proportionally to this time . He further showed
that, conversely, if a similar relation has been verified in the
motion of a body with respect to a certain point, this is
sufficient proof of the action on this body of a force directed constantly towards
this point. This beautiful property is moreover very easily deduced from the
general equations of the curvilinear motion of a molecule, exposed
in the previous lesson, by placing the origin of the coordinates in the center.
forces, and considering the expression of the area described on
any one of the planes coordinated by the corresponding projection of the
vector radius of the mobile. This discovery by Kepler is all the more
remarkable as it took place before the dynamic had actually been
created by Galileo. We will have occasion to notice, in the
astronomical part of this course, that Kepler, having recognized that the
vector rays of the planets describe around the sun areas
proportional to time, which constitutes the first of his three
great astronomical laws, thus concludes that the planets are
continually animated with a tendency towards the sun, of which it was
reserved for Newton to discover the law.
But, whatever the importance of this first area theorem, which
is thus one of the essential bases of celestial mechanics, we should
now see in it only the simplest particular case of the great
general area theorem, discovered almost simultaneously and in
different forms by d'Arcy, Daniel Bernouilli and Euler, around
the middle of the last century. Kepler's discovery was relative
only to the movement of a point: Arcy's discovery relates to the movement of
any system of bodies acting on each other in
any way, which constitutes a case, not only more
complicated, but even essentially different, because of these actions
mutuals. The theorem then consists in that, as a result of these
reciprocal influences, the area which the radius vector
of each molecule of the system will describe separately at each instant around
any point may well be altered, but that the algebraic sum of the
areas thus described by the projections on any plane of
the vector rays of all the molecules, giving each of these
areas the suitable sign according to the ordinary rule, will not suffer any
change, so that, if there is no no other accelerating forces
in the system than these mutual actions, this sum of the areas
described will remain invariable in a given time, and will increase by
therefore in proportion to time. When the system has
no fixed point, this remarkable property takes place relatively to
any point in space; while it is verified only by
taking the fixed point for center of areas, if the system offers one.
Finally, when the bodies of the system are animated by
external accelerating forces , if these forces constantly tend towards the same point, the
areas theorem still subsists, but only with regard to this
point. This last part of the general proposition
obviously provides as a special case, Kepler's theorem, supposing
that the system is reduced to a single molecule.
In the application of this theorem, we usually replace the sum
of the areas corresponding to all the molecules of the system by the
equivalent sum of the products of the mass of each body by the area which
relates to it, which dispenses with sharing the system in molecules of the same
mass.
This is the form in which the general area theorem was
discovered by d'Arcy; this is the one we usually use. As
the area described by the vector radius of each body in an
infinitely small instant is obviously proportional to the product of the
speed of this body by its distance from the fixed point that we are considering, we
can substitute for the sum of the areas the sum of the _momens_ with respect to
this point of all the forces of the system projected on the same
unspecified plane . From this point of view, the areas theorem presents,
according to Laplace's remark, a general property of motion
analogous to one of those of equilibrium, since it then consists in
that this sum of moments, zero in the case of equilibrium, is
constant in the case of movement. This is how this theorem was
found by Euler and by Daniel Bernouilli.
Whatever concrete interpretation one deems appropriate to
give it, it is a simple direct analytical consequence of the formula
general dynamics. To deduce it, it suffices to develop
this formula by forming the equations which relate to the
rotational motion , and in which we will immediately see the
analytical expression of the areas or moments theorem, having regard to the
above conditions. above indicated. From the analytical point of view, we can say
that the utility of this theorem consists essentially in providing in
all cases three first integrals of the general equations of
motion which are in themselves of the second order, which tends to
make the final solution particularly easy. of each
particular dynamic problem .
The area theorem suffices to determine, in the general motion
of any system whatever relates to the motions of
rotation, as the theorem of the center of gravity determines all that
is relative to the motions of translation. Thus, by the only
combination of these two general properties, one could proceed to
the complete study of the motion of any system of bodies, either as
regards translation or as regards rotation.
I must not neglect to point out summarily here, on the subject of
the areas theorem, the unexpected clarity and the admirable simplicity which
M. Poinsot introduced therein by applying to it his fundamental conception
relative to the motions of rotation, which we have considered under the
static point of view in the sixteenth lesson. By substituting for the areas,
or moments considered until then by geometers, the couples
generated by the proposed forces, M. Poinsot made this
theory undergo a very important philosophical improvement, which does
not yet appear to me to have been sufficiently felt. He thus gave a
concrete value, a proper and direct dynamic meaning, to what was
previously only a simple geometric statement of part of the
fundamental equations of motion. Such a happy general transformation
is destined, no doubt, to necessarily increase the resources of
the human mind for the elaboration of dynamic ideas, in all that
concerns the theory of rotational motions. We can see in
M. Poinsot's fine memoir on the properties of moments and areas,
which is annexed to his _Statique_, with what ease he has
succeeded, according to this luminous conception, not only in making
a theory elementary. hitherto founded on the highest analysis,
but to discover in this respect new
very remarkable general properties , which we must not consider here, and which it would have
been difficult to obtain by previous methods.
The areas theorem was, for the illustrious Laplace, the origin of the
discovery of another very remarkable dynamic property, that of this
which he called the _invariable plane_, the consideration of which is especially so
important in celestial mechanics. The sum of the areas projected by
all the bodies of the system on an unspecified plane being constant in a
given time, Laplace sought the direction of the plane with respect to which
this sum was found to be the greatest possible. Now, from the
way in which this plane of the greatest area or of the greatest moment is
determined, Laplace has demonstrated that its direction is necessarily
independent of the mutual reaction of the different parts of the system,
so that, by its nature, this plane must remain continuously
invariable, whatever may be the alterations introduced
in the situation of these bodies by their reciprocal influences, provided
that no new external force arises. We can easily see
what importance must be, as we will explain especially
in the second part of this course, the determination of such a plan
relative to our solar system, since, by relating all our
celestial movements to it, it gives us the 'invaluable advantage of having a
necessarily fixed term of comparison, through all the disturbances
that the mutual action of our planets may cause in the course
of time to their distances, to their revolutions and even to the planes of
their orbits, this which is obviously a first condition
essential so that we can know exactly what
these alterations consist of. Unfortunately we will have occasion to
notice that the uncertainty in which we are so far relative to the
exact value of several essential data, does not
yet allow us to determine with all sufficient precision the situation of
this plane. But this difficulty of application does not in any way affect
the character of this beautiful theorem, considered from the point of view of
rational mechanics, the only one that we should adopt here.
The theory of the invariable plane has been notably perfected in
recent times by M. Poinsot, who naturally had to transport his
proper conception relative to the general theory of areas or
moments. He first considerably simplified the fundamental notion
of this plane, so as to make it as elementary as possible,
by showing that such a plane is really nothing other than the plane of
the general couple resulting from all the couples generated by the
different forces of the system, which immediately defines it by a
very sensitive dynamic property, instead of the only
geometric property of the maximum of areas. When any conception whatever has been
really simplified in its nature, the elaboration being thereby
facilitated, it cannot fail to take on more extension and
lead to new results: such is, in fact, the
ordinary course of the human mind in the sciences, that the theories the
most fertile in discoveries were most often, at their origin,
only a means of rendering simpler the solution of questions already answered
. The work we consider here has offered further
proof of this. Because the theory of M. Poinsot made it possible to introduce a higher
degree of precision in the determination of the invariable plane specific to
our solar system, by pointing out and rectifying an important gap
that Laplace had left there. This great geometer, in calculating the
situation of the plane of the _maximum_ of the areas, thought he should not take
in consideration that the principal areas, produced by the circulation
of the planets around the sun, without taking any account of those due
to the movements of the satellites around the planets, or to the rotation of
all these stars and of the sun itself. M. Poinsot has just proved the
necessity of having regard to these various elements, without which the plan thus
determined could not be regarded as rigorously
invariable; and by seeking the direction of the true invariable plane
as exactly as the actual imperfection of most
of the data implies, he has shown that this plane differs appreciably from that
found by Laplace; what is easy to conceive by the only
consideration of the immense area which the
enormous mass of the sun must introduce into the calculation , although its rotation is very slow.
To complete the indication of the most
important dynamic properties relating to rotational motion, it is now appropriate
to point out here the beautiful theorems discovered by Euler on what he
called the _momens of inertia_ and the _ principal axes_, which we must put
the number of the most important general results of mechanics
rational. Euler gave the name of _moment of inertia_ of an
integral body which expresses the sum of the products of the mass of each
molecule by the square of its distance from the axis around which the body
rotates, an integral the consideration of which must evidently be
very essential, since it may naturally be regarded as the
exact measure of the energy of rotation of the body. When the proposed mass
is homogeneous, this moment of inertia is determined like the other
analogous integrals relating to the shape of a body; when, on the
contrary, this mass is heterogeneous, it is necessary moreover to know the law
of the density in the various layers which compose it, and, apart from that
, the integration is only then only more complicated. This
notion being established, Euler, comparing, in general, the moments of inertia
of the same unspecified body with respect to all the axes of rotation
imaginable passing at a given point, determined the axes relative to
which the moment of inertia must be a _maximum_ or a _minimum_,
especially considering those which intersect at the center of gravity, and which are
distinguished in that they necessarily produce moments less
than if, with the same direction, they were placed everywhere else. He
thus discovered that there are constantly, at any point of a
body, and particularly at the center of gravity, three
rectangular axes , such that the moment of inertia of the body is a _maximum_ with
respect to one of 'between them, and one _minimum_ with respect to another. These
axes are also characterized by another common property which
They are usually used today as an analytical definition, and which
constitutes, in fact, for analysis, the main advantage that we
find in relating the movement of the body to these three axes. This
property consists in that, when these three axes are taken for those
of the coordinates x, y, z, the integrals / int xzdm, / int xydm, / int yzdm
(m expressing the mass of the body), are zero relative to the body all
around, which greatly simplifies the general equations of
rotational motion. But the main dynamic theorem discovered
by Euler with regard to these axes, and according to which he rightly
called them _ principal axes of rotation_, consists in the stability of the
rotations that correspond to them; that is, if the body has
started to rotate around one of these axes, this rotation will persist
indefinitely in the same way, which would not take place for any
other axis, the instantaneous rotation s 'generally running
around a continuously variable axis. This system of principal axes
is generally unique in each body: however, if all moments
of inertia were constantly equal to each other, the direction of these axes would
become totally indeterminate, provided that they are always chosen
perpendicular to each other, which has place, for example, in a
homogeneous sphere , where we can see as permanent axes of rotation
all systems of rectangular axes passing through the center. There
would still be a certain degree of indeterminacy if the body were a
solid of revolution, the geometric axis then being one of the main
dynamic axes ; but the other two can obviously be taken
at will in a plane perpendicular to the first. The determination of the
principal axes often presents great difficulties by considering
bodies of figure and of any constitution; but it is carried out
with extreme facility in uncomplicated cases, which
celestial mechanics happily presents to us as the most common. For example
in a homogeneous ellipsoid, or even only composed of layers
similar and concentric of unequal density, but each of which is
homogeneous, the three conjugated rectangular diameters are themselves
the principal dynamic axes: the moment of inertia of the body is a
_maximum_ relatively to the smallest of these diameters, and a _minimum_ to
respect for the greater. When the principal axes of a body or of a
system are determined as well as the corresponding moments of inertia, if
the system does not revolve around one of these axes, Euler has established
very simple general formulas, which constantly make known the
angles that the line must make with them around which
the instantaneous rotation is spontaneously executed , and the value of the moment of inertia
related to it, which is sufficient for the complete analysis of the
rotational motion .
Such are the general theorems of dynamics which relate
directly to the entire determination of the motion of a body or of
any system, either as regards translation or as regards
rotation. But besides these fundamental properties, geometers have
also discovered several other very general ones which, without being so
strictly indispensable, are singularly deserving of being pointed out in
a philosophical examination of rational mechanics, on account of their
extreme importance for simplification. special searches.
The first and most remarkable of them, that which presents the
most valuable advantages for applications, consists in the famous
theorem of the _conservation of living forces_. The original discovery
is due to Huyghens, who based on this consideration his solution of the
problem of the center of oscillation. The notion was then generalized
by Jean Bernouilli, because Huyghens had established it only in relation to the
movement of heavy bodies. But Jean Bernouilli, granting an
exaggerated and vicious importance to the famous distinction introduced by
Leïbnitz, between the dead forces and the alive forces, tried in
vain to set up this theorem in a primitive law of nature, whereas
it could not be only 'a more or less general consequence of
fundamental dynamic theories. The most important works of which
this property of movement has been the subject are certainly those of
the illustrious Daniel Bernouilli, who gave the live forces theorem its
greatest extension, as well as the systematic form in which
we conceive it today, and who made such a happy use of it especially
for the study of the movement of fluids.
We know that, since Leïbnitz, geometers call _ live force_ of a
body the product of its mass by the square of its speed,
moreover completely disregarding the too vague considerations which
had led Leïbnitz to form such an expression. The theorem
general that we are considering here is that some alterations
which may occur in the motion of each of the bodies of
any system by virtue of their reciprocal action, the sum of the living forces
of all these bodies remains constantly the same in a given time . This is what
we demonstrate today with the greatest ease according to the
fundamental equations of motion of any system, and
above all, as Lagrange did, by starting from the general formula of
the dynamics exposed in the previous lesson. From the
analytical point of view , the extreme utility of this beautiful theorem consists
essentially in that it always provides in advance a first
finite equation between the masses and the speeds of the different bodies of the
system. This relation, which can be considered as one of the
definitive integrals of the differential equations of motion,
suffices for the entire solution of the problem, whenever it is
reducible to the determination of the motion of only one of the bodies that we
consider, a determination which is then carried out with great ease.
But to get a fair idea of this important property, it is
essential to notice that it is subject to a
considerable limitation , which does not allow, with regard to generality, to
place it on the same line as the theorems previously examined.
This limitation, discovered at the end of the last century by Carnot,
consists in the fact that the sum of the living forces constantly undergoes a
decrease in the shock of bodies which are not perfectly
elastic, and generally whenever the system experiences a
sudden change. any. Carnot has shown that then there is a
loss of living forces equal to the sum of the living forces due to the
speeds lost by this change. Thus the theorem of the conservation
of living forces only takes place insofar as the movement of the system varies
only by insensible degrees, or when a shock occurs only between
bodies endowed with perfect elasticity. This important
consideration completes the general notion that one must form of such a
remarkable property.
Of all the great theorems of rational mechanics, the one we
have just considered is without a doubt the most important for
applications to industrial mechanics; that is to say with
regard to the theory of motion of machines, in so far as it is
capable of being established in an exact and precise manner. The theorem
of living forces has hitherto begun to furnish, from this point of view,
very valuable general indications, which have been
presented above all with perfect clarity and conciseness in the work
of Carnot, which we do not has since added nothing really essential. This
theorem presents directly, in fact, the dynamic consideration
of any machine under its true aspect, by showing that,
in any transmission and modification of the movement carried out by a
machine, there is simply an exchange of living force between the mass of the
motor and that of the body to be moved. This exchange would be complete,
that is to say all the living force of the motor would be used while avoiding
sudden changes, if the friction, the resistance of the media,
etc., did not necessarily absorb a more or less
considerable portion according to that the machine is more or less complicated. This
notion brings to light the absurdity of what has been called the
perpetual motion, even indicating in a general way at what
instant the machine abandoned to its only primitive impulse must
spontaneously stop; but this absurdity is, moreover, of its nature
so sensitive, that Huyghens had, on the contrary, partly based his
demonstration of the live forces theorem on the manifest evidence
of such an impossibility. Be that as it may, this theorem gives a
clear idea of the true dynamic perfection of a machine,
reducing it to using the largest possible fraction of the
motor's live force , which can usually only take place in endeavoring to
simplify the mechanism as much as the nature of the engine allows. We
conceives in fact that if we measure, as it seems natural to
do, the useful dynamic effect of an engine in a given time by the
product of the weight that it can raise and the height to which it
transports it , this effect is immediately equivalent, according to the laws of
vertical motion of heavy bodies, to a living force, and not to a
quantity of motion. From this point of view, the famous discussion
raised by Leïbnitz on the subject of living forces, and in which
all the great geometers of this period took part, should not be
regarded as as devoid of reality as d'Alembert seemed to
believe. We were probably mistaken in thinking that the mechanics
rational was interested in this dispute, which in
fact, according to d'Alembert's remark, could not exercise the slightest
real influence over it . The theoretical point of view and the practical point of view
had not been carefully enough separated by the surveyors who
followed this discussion. But, from the point of view of
industrial mechanics alone, it was nonetheless of real
importance. It could even be usefully taken up today, for
the objections which have been made against the vulgar measurement of the
dynamic value of motors deserve serious consideration,
since it does indeed seem unreasonable to take a unit as a unit.
movement which is not uniform.
But, whatever decision one ends up adopting on this
unfinished dispute , the application of the theorem of the living forces will nevertheless retain
all its importance in showing in its true light the
real destination of machines, by proving that necessarily they
lose in speed or time what they gain in strength or
vice versa, so that their usefulness consists essentially
in exchanging the various factors of the effect to be
produced, without ever being able to increase it. by themselves in its
totality, and by constantly subjecting it to a constant
inevitable reduction, usually very notable. It is doubtful,
moreover, that the application of this theorem can at any time be
pushed much further than general indications of this kind,
because the true calculation _a priori_ of the precise effect of
any given machine presents, as a problem of dynamics, too great a
complication, and requires the exact knowledge of too many
still completely unknown relations, to be able to be effectively
attempted in most cases [29].
[Note 29: The true proper theory of
industrial mechanics , which is by no means, as is
often believed , a simple derivation of _phoronomy_ or
rational mechanics, and which relates to a
completely distinct order of ideas , has not yet been conceived. It
is, in this respect, as with any other _science
of application_ of which the human mind has hitherto only a
few insufficient elements, according to the remark indicated
in our second lesson. Industrial mechanics,
apart from the formation of motors, which depends on
the totality of our knowledge of nature, is composed of
two very different classes of research, some
dynamic, others geometric. The first are
aimed at determining the most suitable devices,
in order to use the
given driving forces as much as possible ; that is to say to obtain between the living force of the body
to be moved and that of the motor the closest relation to
the unit, having regard to the modifications required in the
speed by the known destination of the machine. As for the
others, it is proposed to change at will, using a
suitable mechanism, the lines described by the points
of application of the forces. In short, the movement is
modified, in some, as to its intensity; in
others, as to its direction. The former relate to
an entirely new doctrine, on the subject of which he
yet no direct and truly
rational design has been produced . It is more or less the same for the others,
which depend on this _ situational
geometry_ glimpsed by Leïbnitz, but which has hitherto made almost no progress.
In this regard, I know of no other real work than an
ingenious elementary consideration presented by Monge, and
which, although simply empirical, deserves to be noted here,
if only to indicate the true nature of this order.
of ideas.
Monge started from this observation, very plausible in
fact, that, in reality, the movements executed by
machines are either rectilinear or circular, each one
being able to be moreover or continuous or alternating. He therefore
considered every machine as destined, under the
geometrical relation , to transform these various elementary motions
into one another. That being said, by exhausting all the
various combinations that such a transformation can
offer, it has necessarily seen the result of ten series
of devices in which all
known machines can be stored , as well as those which will be imagined
later. The tables resulting from this classification can
therefore be considered as showing the mechanic the
empirical means of solving, in each case, the problem
of the transformation of motion, by choosing, among
all the devices suitable for fulfilling the proposed condition, the
one which moreover presents the most advantages.]
The motion of a system any one presents another
very remarkable general property , although less important, either in the
analytical relation, or especially in the physical relation, than that which
has just been examined: it is the property expressed by the famous
general theorem of dynamics to which Maupertuis has given the
so vicious denomination of _principle of the least action_.
The lineage of ideas about this discovery goes back to a time
very remote, because the geometers of antiquity had already made
some remarks which can be conceived today as equivalent
to the verification of this theorem in the simplest particular case.
Ptolemy, in fact, expressly observes, with regard to the law of the reflection
of light, that by the nature of this law, the light by
reflecting itself finds itself following the shortest possible path to
reach from one point to another. When Descartes and Snellius
discovered the real law of refraction, Fermat investigated whether this
could not be achieved _a priori_ from some consideration
analogous to Ptolemy's remark. The _minimum_ cannot then have
place relative to the length of the path traveled, since the
straight road would have been possible in this case, Fermat presumed that it would exist
with respect to time. He therefore proposed to himself, looking at the route of
light as composed of two different straight lines, separated, at an
unknown angle, on the surface of the refractive body, what must be
this relative direction so that the time used by the light in
its path was the least possible, and he was fortunate to find
from this consideration alone a law of refraction exactly in
conformity with that directly deduced from the observations by Snellius and
Descartes. This beautiful solution is also eminently
remarkable in the general history of the progress of
mathematical analysis , as having offered Fermat the first
important application of his famous method _de maximis et minimis_, which contains
the true primitive germ of differential calculus.
The comparison of Ptolemy's remark with Fermat's work
considered from the dynamic point of view, became for Maupertuis the basis
for the discovery of the theorem we are considering. Although lost, much
more than led, by vague metaphysical considerations on the
supposed economy of forces in nature, he ends up arriving at this
important result, that the trajectory of a body subjected to the action of
Any force must necessarily be such that the integral of the
product of the speed of the moving body by the element of the curve described was
always a _minimum_, relative to its value in any other curve.
But Lagrange is with justice generally regarded by the
current geometers like the true founder of this theorem, not-only for
having generalized it as much as possible, but especially for having
discovered the true demonstration of it by relating it to the
fundamental dynamic theories , and in freeing him from the confused and
arbitrary notions that Maupertuis had used. There now remains no
other trace of Maupertuis' work than the name he imposed on this
theorem, and whose impropriety is universally recognized, although,
for the sake of brevity, it has continued to be used. The theorem, as
it was established by Lagrange with respect to any system of
bodies, consists in that, whatever their
reciprocal attractions , or their tendencies towards fixed centers, the trajectories
described by these bodies are always such that the sum of the products of
the mass of each of them, and of the integral relating to its speed
multiplied by the element of the corresponding curve, is necessarily
a _maximum_ or a _minimum_, this sum being extended to the whole of the
system. It is also important to note that the demonstration of this
General theorem being based on the theorem of the living forces, it is
inevitably subject to the same limitations as the latter.
Besides the beautiful property of motion contained in this
remarkable proposition , we can see that, from the analytical point of view, it can be
considered as a new means of forming the differential equations
which must lead to the determination of each special motion. It
suffices, in fact, in accordance with the general method of _maxima_ and
_minima_ provided by the calculation of the variations, to express that the sum
previously indicated is a _maximum_ or a _minimum_ (either absolute
or relative depending on the case), by making its zero variation. Lagrange has
expressly shown how, from this consideration alone, we can,
in general, find the fundamental formula of dynamics. But, however
useful such a way of
proceeding may be in certain cases , its importance should not be exaggerated; for we must not
lose sight of the fact that it does not by itself provide any finite integral
of the equations of motion; it confines itself only to establishing these
equations in another way, which may sometimes be more
suitable. In this respect, the theorem of least action is
certainly less valuable than that of living forces. Anyway
, it should be noted here with Lagrange that all of these
two theorems can be regarded, in general thesis, as sufficient
for the entire determination of the motion of a body.
The theorem of the least action was also presented by Lagrange in
another general form, especially intended to make more sensitive
its concrete interpretation. Indeed, the element of the trajectory
can obviously be replaced in the statement of this theorem by the
equivalent product of the speed and the element of time, the theorem
then consists in that each body of the system constantly describes a
curve such that the sum of the living forces consumed in a given time
to reach from one position to another is necessarily a _maximum_
or a _minimum_.
The philosophical history of the work relating to the theorem of the least
action is particularly apt to bring to light
the complete insufficiency and the radical vice of the
metaphysical considerations employed as means of scientific discovery. It
can not be denied, without doubt, that the theological and metaphysical principle of
final causes did not have some use here,
initially contributing to awakening the attention of geometers to this important
dynamic property, and even providing them in this regard some
vague indications. The spirit of this course, as we have already
expressly pointed out, and as it will develop more and more by the
continuation, in fact, prescribes us to regard, as a general thesis,
theological and metaphysical hypotheses as having been useful and even
necessary for the real progress of human intelligence, by sustaining
its activity as long as absence of
positive conceptions of sufficient generality. But even then, the many
fundamental drawbacks inherent in such a way of proceeding
clearly verify that it can only be seen as
provisional. The current example offers significant proof of this. For, without
the introduction of the exact and real considerations founded on the
general laws of mechanics, one would still dispute, as well as the
remarks Lagrange with so much reason, on what is meant by
_the least action_ of nature, the alleged economy of forces
consisting sometimes in space, sometimes in time, and most
often not in fact neither. It is moreover evident
that this property does not have that absolute character which one had at first
wished to impose on it, since it experiences in a large number of cases
determined restrictions. But what above all makes manifest the
radical vice of the primitive considerations is that, according to the
exact analysis of the question treated by Lagrange, we see that the integral
defined above n '
_minimum_, and that it can, on the contrary, be just as well a
_maximum_, as it actually happens in certain cases, the true
general theorem consisting only in that the variation of this
integral is zero: what then becomes of the_economy_ of forces, in
whatever way one claims to characterize the_action_? The insufficiency
and even the error of Maupertuis's argument are therefore
fully evident. On this occasion, as on all those on which
there has been a competition so far, the comparison has expressly established
the immense and necessary superiority of positive
philosophy over theological and metaphysical philosophy, not only as regards
correctness and precision of the actual results, but even as to
the breadth of conceptions and the real elevation from the
intellectual point of view .
To complete this reasoned enumeration of the general properties of
motion, I believe I should finally point out here a last
very remarkable proposition , which is not ordinarily placed in the same
category as the preceding ones, and which nevertheless deserves, to such a high
degree, of to fix our attention, either by its intrinsic beauty, or
especially by the importance and the extent of its applications to
the most difficult dynamic problems . This is the famous general theorem
discovered by Daniel Bernouilli, on the _coexistence of small
oscillations_. Here is what it consists of.
We saw, starting this lesson, that there exists, for any
system of forces, a situation of equilibrium _stable_, that in
which the sum of the living forces is one of the _maximum_, according to the law
of Maupertuis generalized by Lagrange. When the system is infinitely
little removed from this situation by any cause whatsoever, it tends to
return to it, making around it a series of infinitely
small oscillations , gradually diminished and soon destroyed by the resistance
of the environment and friction, and that can be compared to those of a
pendulum of a suitable length subjected to the influence of gravity
determined. But several different causes can
simultaneously cause the system to oscillate in various ways around the
position of stability. That said, Daniel Bernouilli's theorem
consists in the fact that all the species of infinitely small oscillations
produced by these various simultaneous disturbances, whatever their
nature, simply do nothing but superimpose themselves, by coexisting without
harming each other, each of them. taking place as if she were alone. It is
easy to see the extreme importance of this fine proposition in facilitating
the study of such a kind of motion, since it suffices from that
to analyze separately each kind of oscillations produced by each
separate disturbance. This decomposition is especially of the greatest
utility in research relating to the motion of fluids, where
such an order of considerations arises almost constantly. But the
property discovered by Daniel Bernouilli is no less interesting
from a physical point of view than from a logical point of view. In fact,
considered as a law of nature, it explains directly, in the
most satisfactory manner, a multitude of various facts, which
observation had long noted, and which we had tried in
vain to conceive until then. . Such is, for example, the
coexistence of waves produced on the surface of a liquid, when
is agitated at the same time in several different points by various
causes whatever. Such, above all, in acoustics, is the
simultaneity of the distinct sounds produced by various movements of
the air. This coexistence, which takes place without confusion between the
different sound waves, had obviously been often observed,
since it is one of the essential bases of the mechanism of our
hearing; but it seemed inexplicable; we now see
only an immediate consequence of Daniel Bernouilli's beautiful theorem.
Considering this theorem from the most philosophical point of view,
we do not perhaps find it less remarkable for the way in which it
results from the general equations of motion, only by its
analytical or physical importance . Indeed this coexistence of the various orders
of infinitely small oscillations of any system, around its
situation of stability, takes place because the differential equation which
expresses the law of any one of these motions happens to be
_linear_, and consequently from the class of those whose
general integral is necessarily the simple sum of a certain number
of particular integrals. Thus, under the analytical report, the
superposition of the various oscillatory motions has for cause the kind
of superposition which is then established between the various integrals
corresponding. This important correlation is certainly, as
Laplace rightly observes, one of the finest examples of this
necessary harmony between the abstract and the concrete, of which
mathematical philosophy has offered us so many admirable verifications.
These are the main philosophical considerations relating to the
various general theorems discovered so far in
rational mechanics , and which all derive, as simple
analytical deductions more or less distant, from the fundamental laws of motion
on which the entire system of phoronomic science is based. .
The summary examination of these theorems, the whole of which constitutes one of the
the most imposing monuments of the activity of the
properly directed human intelligence , was indispensable to complete the determination of
the philosophical character of the science of balance and movement,
already sufficiently traced in the preceding lessons, with regard to the
method. We can now therefore form a clear
general idea of the proper nature of this second branch of
concrete mathematics , which was to be the sole essential object of our work on
this subject.
I have endeavored, in this volume, to make one feel, as far as was
in my power, of what
mathematical philosophy really consists , either as regards its abstract conceptions or as regards its
various orders of concrete considerations, or finally with regard to the
intimate and permanent correlation which necessarily exists between
them. I deeply regret that the limits within which
I had to confine myself, given the purpose of this work, did not
allow me to convey, as much as I would have wished, in the
reader's mind my deep feeling of the nature of this immense and admirable
science, which, the necessary basis of the whole of positive philosophy,
constitutes moreover evidently, in itself, the most
indisputable testimony to the scope of human genius. But I hope that thinkers
who are not unfortunate enough to be entirely foreign to this science
fundamental will be able, according to the reflections I have indicated, to
arrive at a clear conception of its true philosophical character.
To present a truly comprehensive overview of mathematical philosophy
in its current state, I have indicated in advance (see Lesson 3) that
I still have to consider a third branch of
concrete mathematics , that which consists in the application of analysis to the study
of thermological phenomena, the last great conquest of the
human mind , due to the illustrious friend whose recent loss I deplore,
the immortal Fourier, who has just left in the learned world so
deep gap, long destined to be day by day more
strongly felt. But, in order not to deviate as much as possible from the
habits still universally adopted, I announced that I thought I should
postpone this important examination until the natural order of the
considerations set out in this work led us to part
of physics that deals with thermology. Although such a
transposition is not really rational, it can only
result from it a secondary inconvenience, the
philosophical appreciation which I will present having moreover exactly the same
character as if it had been placed in its true logical rank. .
Now considering mathematical philosophy as
fully characterized, we must proceed to the examination of its
more or less perfect application to the study of the various orders of
natural phenomena according to their degree of simplicity, an application which,
by itself, is moreover evidently suitable to throw a new light
on the true principles of this philosophy, and without which, in
fact, they could not be properly appreciated. Such will be the object
of the following volume, while conforming to the encyclopedic order
rigorously determined in the second lesson, according to the
special nature of each of the principal classes of phenomena which we have
established, and, consequently, beginning with phenomenons
astronomical studies for which mathematical science is
eminently destined.
END OF TOME ONE.
TABLE OF CONTENTS
CONTAINED IN TOME ONE.
Dedication v
Warning from the author.
Synoptic table of the whole course of positive philosophy.
1st LESSON .-- Statement of the purpose of this course, or general considerations
on the nature and importance of positive philosophy.
2nd LESSON .-- Exposure of the outline of this course, or general considerations
on the hierarchy of positive sciences.
LESSON 3 .-- Philosophical considerations on the whole of
mathematical science .
LESSON 4 - General view of mathematical analysis.
5th LESSON .-- General considerations on the computation of
direct functions .
6th LESSON .-- Comparative exposition of the various general points of view
from which the calculation of indirect functions can be considered.
7th LESSON .-- General table of the calculation of indirect functions.
8th LESSON .-- General considerations on the calculation of variations.
9th LESSON .-- General considerations on
finite difference calculus .
10th LESSON .-- General view of geometry.
11th LESSON .-- General considerations on _special_ or
_preliminary_ geometry .
12th LESSON .-- Fundamental conception of _general_ or
_analytical_ geometry .
13th LESSON .-- Two-dimensional _general_ geometry.
14th LESSON .-- Of _general_ geometry in three dimensions.
LESSON 15 .-- Philosophical Considerations on the
Fundamentals of Rational Mechanics.
16th LESSON .-- General view of statics.
17th LESSON .-- General view of dynamics.
18th LESSON .-- Considerations on general theorems of
rational mechanics .
ERRATA OF THE FIRST VOLUME.
TRANSCRIBER'S NOTE: These errors have been corrected in this
document. The list is reproduced here only for
reference.
Page 147, line 25, _instead of_ ideas, _read_ designs.
201 1 fact, _read_ knows.
236 12 M. Fournier, _read_ M. Fourier.
248 26 _remove_ or less.
351 17 _after_ influence, _add_ singularly.
420 11 _instead of_ signs, _read_ lines.
469 1 so far, _read_ till date.
504 20 intensity, _read_ intimacy.
508 18, _after_ volume of, _add_ truncated cone or.
509 3, _instead of_ S = 2 / pi / int ydx, _read_ S = 2 / pi / int yds.
530 20 various individuals, _read_ various species.
534 last line of the note, _before_ denoting, _add_.
556 6 _instead of_ relationships, _read_ actions.
624 4 operations, _read_ equations.
