ORDINARY ANALYSIS, OR ALGEBRA.
The _Calculus of direct Functions_, or _Algebra_, is (as was shown at
the end of the preceding chapter) entirely sufficient for the solution
of mathematical questions, when they are so simple that we can form
directly the equations between the magnitudes themselves which we are
considering, without its being necessary to introduce in their place, or
conjointly with them, any system of auxiliary quantities _derived_ from
the first. It is true that in the greatest number of important cases its
use requires to be preceded and prepared by that of the _Calculus of
indirect Functions_, which is intended to facilitate the establishment
of equations. But, although algebra has then only a secondary office to
perform, it has none the less 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 base of all mathematical
analysis. We must therefore, before going any further, consider in a
general manner the logical composition of this calculus, and the degree
of development to which it has at the present day arrived.
_Its Object._ The final object of this calculus being the _resolution_
(properly so called) of _equations_, that is, the discovery of the
manner in which the unknown quantities are formed from the known
quantities, in accordance with the _equations_ which exist between them,
it naturally presents as many different departments as we can conceive
truly distinct classes of equations. Its appropriate extent is
consequently rigorously indefinite, the number of analytical functions
susceptible of entering into equations being in itself quite unlimited,
although they are composed of only a very small number of primitive
elements.
_Classification of Equations._ The rational classification of equations
must evidently be determined by the nature of the analytical elements of
which their numbers are composed; every other classification would be
essentially arbitrary. Accordingly, analysts begin by dividing equations
with one or more variables into two principal classes, according as they
contain functions of only the first three couples (see the table in
chapter i., page 51), or as they include also exponential or circular
functions. The names of _Algebraic_ functions and _Transcendental_
functions, commonly given to these two principal groups of analytical
elements, are undoubtedly very inappropriate. But the universally
established division between the corresponding equations is none the
less very real in this sense, that the resolution of equations
containing the functions called _transcendental_ necessarily presents
more difficulties than those of the equations called _algebraic_. Hence
the study of the former is as yet exceedingly imperfect, so that
frequently the resolution of the most simple of them is still unknown to
us,[7] and our analytical methods have almost exclusive reference to the
elaboration of the latter.
[Footnote 7: Simple as may seem, for example, the equation
_a^x_ + _b^x_ = _c^x_,
we do not yet know how to resolve it, which may give some idea of
the extreme imperfection of this part of algebra.]
ALGEBRAIC EQUATIONS.
Considering now only these _Algebraic_ equations, we must observe, in
the first place, that although they may often contain _irrational_
functions of the unknown quantities as well as _rational_ functions, we
can always, by more or less easy transformations, make the first case
come under the second, so that it is with this last that analysts have
had to occupy themselves exclusively in order to resolve all sorts of
_algebraic_ equations.
_Their Classification._ In the infancy of algebra, these equations were
classed according to the number of their terms. But this classification
was evidently faulty, since it separated cases which were really
similar, and brought together others which had nothing in common besides
this unimportant characteristic.[8] It has been retained only for
equations with two terms, which are, in fact, capable of being resolved
in a manner peculiar to themselves.
[Footnote 8: The same error was afterward committed, in the infancy
of the infinitesimal calculus, in relation to the integration of
differential equations.]
The classification of equations by what is called their _degrees_, is,
on the other hand, eminently natural, for this distinction rigorously
determines the greater or less difficulty of their _resolution_. This
gradation is apparent in the cases of all the equations which can be
resolved; but it may be indicated in a general manner independently of
the fact of the resolution. We need only consider that the most general
equation of each degree necessarily comprehends all those of the
different inferior degrees, as must also the formula which determines
the unknown quantity. Consequently, however slight we may suppose the
difficulty peculiar to the _degree_ which we are considering, since it
is inevitably complicated in the execution with those presented by all
the preceding degrees, the resolution really offers more and more
obstacles, in proportion as the degree of the equation is elevated.
ALGEBRAIC RESOLUTION OF EQUATIONS.
_Its Limits._ The resolution of algebraic equations is as yet known to
us only in the four first degrees, such is the increase of difficulty
noticed above. In this respect, algebra has made no considerable
progress since the labours of Descartes and the Italian analysts of the
sixteenth century, although in the last two centuries there has been
perhaps scarcely a single geometer who has not busied himself in trying
to advance the resolution of equations. The general equation of the
fifth degree itself has thus far resisted all attacks.
The constantly increasing complication which the formulas for resolving
equations must necessarily present, in proportion as the degree
increases (the difficulty of using the formula of the fourth degree
rendering it almost inapplicable), has determined analysts to renounce,
by a tacit agreement, the pursuit of such researches, although they are
far from regarding it as impossible to obtain the resolution of
equations of the fifth degree, and of several other higher ones.
_General Solution._ The only question of this kind which would be really
of great importance, at least in its logical relations, would be the
general resolution of algebraic equations of any degree whatsoever. Now,
the more we meditate on this subject, the more we are led to think, with
Lagrange, that it really surpasses the scope of our intelligence. We
must besides observe that the formula which would express the _root_ of
an equation of the _m^{th}_ degree would necessarily include radicals of
the _m^{th}_ order (or functions of an equivalent multiplicity), because
of the _m_ determinations which it must admit. Since we have seen,
besides, that this formula must also embrace, as a particular case, that
formula which corresponds to every lower degree, it follows that it
would inevitably also contain radicals of the next lower degree, the
next lower to that, &c., so that, even if it were possible to discover
it, it would almost always present too great a complication to be
capable of being usefully employed, unless we could succeed in
simplifying it, at the same time retaining all its generality, by the
introduction of a new class of analytical elements of which we yet have
no idea. We have, then, reason to believe that, without having already
here arrived at the limits imposed by the feeble extent of our
intelligence, we should not be long in reaching them if we actively and
earnestly prolonged this series of investigations.
It is, besides, important to observe that, even supposing we had
obtained the resolution of _algebraic_ equations of any degree whatever,
we would still have treated only a very small part of _algebra_,
properly so called, that is, of the calculus of direct functions,
including the resolution of all the equations which can be formed by the
known analytical functions.
Finally, we must remember that, by an undeniable law of human nature,
our means for conceiving new questions being much more powerful than our
resources for resolving them, or, in other words, the human mind being
much more ready to inquire than to reason, we shall necessarily always
remain _below_ the difficulty, no matter to what degree of development
our intellectual labour may arrive. Thus, even though we should some day
discover the complete resolution of all the analytical equations at
present known, chimerical as the supposition is, there can be no doubt
that, before attaining this end, and probably even as a subsidiary
means, we would have already overcome the difficulty (a much smaller
one, though still very great) of conceiving new analytical elements, the
introduction of which would give rise to classes of equations of which,
at present, we are completely ignorant; so that a similar imperfection
in algebraic science would be continually reproduced, in spite of the
real and very important increase of the absolute mass of our knowledge.
_What we know in Algebra._ In the present condition of algebra, the
complete resolution of the equations of the first four degrees, of any
binomial equations, of certain particular equations of the higher
degrees, and of a very small number of exponential, logarithmic, or
circular equations, constitute the fundamental methods which are
presented by the calculus of direct functions for the solution of
mathematical problems. But, limited as these elements are, geometers
have nevertheless succeeded in treating, in a truly admirable manner, a
very great number of important questions, as we shall find in the course
of the volume. The general improvements introduced within a century into
the total system of mathematical analysis, have had for their principal
object to make immeasurably useful this little knowledge which we have,
instead of tending to increase it. This result has been so fully
obtained, that most frequently this calculus has no real share in the
complete solution of the question, except by its most simple parts;
those which have reference to equations of the two first degrees, with
one or more variables.
NUMERICAL RESOLUTION OF EQUATIONS.
The extreme imperfection of algebra, with respect to the resolution of
equations, has led analysts to occupy themselves with a new class of
questions, whose true character should be here noted. They have busied
themselves in filling up the immense gap in the resolution of algebraic
equations of the higher degrees, by what they have named the _numerical
resolution_ of equations. Not being able to obtain, in general, the
_formula_ which expresses what explicit function of the given quantities
the unknown one is, they have sought (in the absence of this kind of
resolution, the only one really _algebraic_) to determine, independently
of that formula, at least the _value_ of each unknown quantity, for
various designated systems of particular values attributed to the given
quantities. By the successive labours of analysts, this incomplete and
illegitimate operation, which presents an intimate mixture of truly
algebraic questions with others which are purely arithmetical, has been
rendered possible in all cases for equations of any degree and even of
any form. The methods for this which we now possess are sufficiently
general, although the calculations to which they lead are often so
complicated as to render it almost impossible to execute them. We have
nothing else to do, then, in this part of algebra, but to simplify the
methods sufficiently to render them regularly applicable, which we may
hope hereafter to effect. In this condition of the calculus of direct
functions, we endeavour, in its application, so to dispose the proposed
questions as finally to require only this numerical resolution of the
equations.
_Its limited Usefulness._ Valuable as is such a resource in the absence
of the veritable solution, it is essential not to misconceive the true
character of these methods, which analysts rightly regard as a very
imperfect algebra. In fact, we are far from being always able to reduce
our mathematical questions to depend finally upon only the _numerical_
resolution of equations; that can be done only for questions quite
isolated or truly final, that is, for the smallest number. Most
questions, in fact, are only preparatory, and intended to serve as an
indispensable preparation for the solution of other questions. Now, for
such an object, it is evident that it is not the actual _value_ of the
unknown quantity which it is important to discover, but the _formula_,
which shows how it is derived from the other quantities under
consideration. It is this which happens, for example, in a very
extensive class of cases, whenever a certain question includes at the
same time several unknown quantities. We have then, first of all, to
separate them. By suitably employing the simple and general method so
happily invented by analysts, and which consists in referring all the
other unknown quantities to one of them, the difficulty would always
disappear if we knew how to obtain the algebraic resolution of the
equations under consideration, while the _numerical_ solution would then
be perfectly useless. It is only for want of knowing the _algebraic_
resolution of equations with a single unknown quantity, that we are
obliged to treat _Elimination_ as a distinct question, which forms one
of the greatest special difficulties of common algebra. Laborious as are
the methods by the aid of which we overcome this difficulty, they are
not even applicable, in an entirely general manner, to the elimination
of one unknown quantity between two equations of any form whatever.
In the most simple questions, and when we have really to resolve only a
single equation with a single unknown quantity, this _numerical_
resolution is none the less a very imperfect method, even when it is
strictly sufficient. It presents, in fact, this serious inconvenience of
obliging us to repeat the whole series of operations for the slightest
change which may take place in a single one of the quantities
considered, although their relations to one another remain unchanged;
the calculations made for one case not enabling us to dispense with any
of those which relate to a case very slightly different. This happens
because of our inability to abstract and treat separately that purely
algebraic part of the question which is common to all the cases which
result from the mere variation of the given numbers.
According to the preceding considerations, the calculus of direct
functions, viewed in its present state, divides into two very distinct
branches, according as its subject is the _algebraic_ resolution of
equations or their _numerical_ resolution. The first department, the
only one truly satisfactory, is unhappily very limited, and will
probably always remain so; the second, too often insufficient, has, at
least, the advantage of a much greater generality. The necessity of
clearly distinguishing these two parts is evident, because of the
essentially different object proposed in each, and consequently the
peculiar point of view under which quantities are therein considered.
_Different Divisions of the two Methods of Resolution._ If, moreover, we
consider these parts with reference to the different methods of which
each is composed, we find in their logical distribution an entirely
different arrangement. In fact, the first part must be divided according
to the nature of the equations which we are able to resolve, and
independently of every consideration relative to the _values_ of the
unknown quantities. In the second part, on the contrary, it is not
according to the _degrees_ of the equations that the methods are
naturally distinguished, since they are applicable to equations of any
degree whatever; it is according to the numerical character of the
_values_ of the unknown quantities; for, in calculating these numbers
directly, without deducing them from general formulas, different means
would evidently be employed when the numbers are not susceptible of
having their values determined otherwise than by a series of
approximations, always incomplete, or when they can be obtained with
entire exactness. This distinction of _incommensurable_ and of
_commensurable_ roots, which require quite different principles for
their determination, important as it is in the numerical resolution of
equations, is entirely insignificant in the algebraic resolution, in
which the _rational_ or _irrational_ nature of the numbers which are
obtained is a mere accident of the calculation, which cannot exercise
any influence over the methods employed; it is, in a word, a simple
arithmetical consideration. We may say as much, though in a less degree,
of the division of the commensurable roots themselves into _entire_ and
_fractional_. In fine, the case is the same, in a still greater degree,
with the most general classification of roots, as _real_ and
_imaginary_. All these different considerations, which are preponderant
as to the numerical resolution of equations, and which are of no
importance in their algebraic resolution, render more and more sensible
the essentially distinct nature of these two principal parts of algebra.
THE THEORY OF EQUATIONS.
These two departments, which constitute the immediate object of the
calculus of direct functions, are subordinate to a third one, purely
speculative, from which both of them borrow their most powerful
resources, and which has been very exactly designated by the general
name of _Theory of Equations_, although it as yet relates only to
_Algebraic_ equations. The numerical resolution of equations, because of
its generality, has special need of this rational foundation.
This last and important branch of algebra is naturally divided into two
orders of questions, viz., those which refer to the _composition_ of
equations, and those which concern their _transformation_; these latter
having for their object to modify the roots of an equation without
knowing them, in accordance with any given law, providing that this law
is uniform in relation to all the parts.[9]
[Footnote 9: The fundamental principle on which reposes the theory
of equations, and which is so frequently applied in all
mathematical analysis--the decomposition of algebraic, rational,
and entire functions, of any degree whatever, into factors of the
first degree--is never employed except for functions of a single
variable, without any one having examined if it ought to be
extended to functions of several variables. The general
impossibility of such a decomposition is demonstrated by the author
in detail, but more properly belongs to a special treatise.]
THE METHOD OF INDETERMINATE COEFFICIENTS.
To complete this rapid general enumeration of the different essential
parts of the calculus of direct functions, I must, lastly, mention
expressly one of the most fruitful and important theories of algebra
proper, that relating to the transformation of functions into series by
the aid of 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
some of its importance since the invention and the development of the
infinitesimal calculus, the place of which it might so happily take in
some particular respects. But the increasing extension of the
transcendental analysis, although it has rendered this method much less
necessary, has, on the other hand, multiplied its applications and
enlarged its resources; so that by the useful combination between the
two theories, which has finally been effected, the use of the method of
indeterminate coefficients has become at present much more extensive
than it was even before the formation of the calculus of indirect
functions.
* * * * *
Having thus sketched the general outlines of algebra proper, I have now
to offer some considerations on several leading points in the calculus
of direct functions, our ideas of which may be advantageously made more
clear by a philosophical examination.
IMAGINARY QUANTITIES.
The difficulties connected with several peculiar symbols to which
algebraic calculations sometimes lead, and especially to the expressions
called _imaginary_, have been, I think, much exaggerated through purely
metaphysical considerations, which have been forced upon them, in the
place of regarding these abnormal results in their true point of view as
simple analytical facts. Viewing them thus, we readily see that, since
the spirit of mathematical analysis consists in considering magnitudes
in reference to their relations only, and without any regard to their
determinate value, analysts are obliged to admit indifferently every
kind of expression which can be engendered by algebraic combinations.
The interdiction of even one expression because of its apparent
singularity would destroy the generality of their conceptions. The
common embarrassment on this subject seems to me to proceed essentially
from an unconscious confusion between the idea of _function_ and the
idea of _value_, or, what comes to the same thing, between the
_algebraic_ and the _arithmetical_ point of view. A thorough examination
would show mathematical analysis to be much more clear in its nature
than even mathematicians commonly suppose.
NEGATIVE QUANTITIES.
As to negative quantities, which have given rise to so many misplaced
discussions, as irrational as useless, we must distinguish between their
_abstract_ signification and their _concrete_ interpretation, which have
been almost always confounded up to the present day. Under the first
point of view, the theory of negative quantities can be established in a
complete manner by a single algebraical consideration. The necessity of
admitting such expressions is the same as for imaginary quantities, as
above indicated; and their employment as an analytical artifice, to
render the formulas more comprehensive, is a mechanism of calculation
which cannot really give rise to any serious difficulty. We may
therefore regard the abstract theory of negative quantities as leaving
nothing essential to desire; it presents no obstacles but those
inappropriately introduced by sophistical considerations.
It is far from being so, however, with their concrete theory. This
consists essentially in that admirable property of the signs + and-, of
representing analytically the oppositions of directions of which certain
magnitudes are susceptible. This _general theorem_ on the relation of
the concrete to the abstract in mathematics is one of the most beautiful
discoveries which we owe to the genius of Descartes, who obtained it as
a simple result of properly directed philosophical observation. A great
number of geometers have since striven to establish directly its general
demonstration, but thus far their efforts have been illusory. Their vain
metaphysical considerations and heterogeneous minglings of the abstract
and the concrete have so confused the subject, that it becomes necessary
to here distinctly enunciate the general fact. It consists in this: if,
in any equation whatever, expressing the relation of certain quantities
which are susceptible of opposition of directions, one or more of those
quantities come to be reckoned in a direction contrary to that which
belonged to them when the equation was first established, it will not be
necessary to form directly a new equation for this second state of the
phenomena; it will suffice to change, in the first equation, the sign of
each of the quantities which shall have changed its direction; and the
equation, thus modified, will always rigorously coincide with that which
we would have arrived at in recommencing to investigate, for this new
case, the analytical law of the phenomenon. The general theorem consists
in this constant and necessary coincidence. Now, as yet, no one has
succeeded in directly proving this; we have assured ourselves of it only
by a great number of geometrical and mechanical verifications, which
are, it is true, sufficiently multiplied, and especially sufficiently
varied, to prevent any clear mind from having the least doubt of the
exactitude and the generality of this essential property, but which, in
a philosophical point of view, do not at all dispense with the research
for so important an explanation. The extreme extent of the theorem must
make us comprehend both the fundamental difficulties of this research
and the high utility for the perfecting of mathematical science which
would belong to the general conception of this great truth. This
imperfection of theory, however, has not prevented geometers from making
the most extensive and the most important use of this property in all
parts of concrete mathematics.
It follows from the above general enunciation of the fact, independently
of any demonstration, that the property of which we speak must never be
applied to magnitudes whose directions are continually varying, without
giving rise to a simple opposition of direction; in that case, the sign
with which every result of calculation is necessarily affected is not
susceptible of any concrete interpretation, and the attempts sometimes
made to establish one are erroneous. This circumstance occurs, among
other occasions, in the case of a radius vector in geometry, and
diverging forces in mechanics.
PRINCIPLE OF HOMOGENEITY.
A second general theorem on the relation of the concrete to the abstract
is that which is ordinarily designated under the name of _Principle of
Homogeneity_. It is undoubtedly much less important in its applications
than the preceding, but it particularly merits our attention as having,
by its nature, a still greater extent, since it is applicable to all
phenomena without distinction, and because of the real utility which it
often possesses for the verification of their analytical laws. I can,
moreover, exhibit a direct and general demonstration of it which seems
to me very simple. It is founded on this single observation, which is
self-evident, that the exactitude of every relation between any concrete
magnitudes whatsoever is independent of the value of the _units_ to
which they are referred for the purpose of expressing them in numbers.
For example, the relation which exists between the three sides of a
right-angled triangle is the same, whether they are measured by yards,
or by miles, or by inches.
It follows from this general consideration, that every equation which
expresses the analytical law of any phenomenon must possess this
property of being in no way altered, when all the quantities which are
found in it are made to undergo simultaneously the change corresponding
to that which their respective units would experience. Now this change
evidently consists in all the quantities of each sort becoming at once
_m_ times smaller, if the unit which corresponds to them becomes _m_
times greater, or reciprocally. Thus every equation which represents any
concrete relation whatever must possess this characteristic of remaining
the same, when we make _m_ times greater all the quantities which it
contains, and which express the magnitudes between which the relation
exists; excepting always the numbers which designate simply the mutual
_ratios_ of these different magnitudes, and which therefore remain
invariable during the change of the units. It is this property which
constitutes the law of Homogeneity in its most extended signification,
that is, of whatever analytical functions the equations may be composed.
But most frequently we consider only the cases in which the functions
are such as are called _algebraic_, and to which the idea of _degree_ is
applicable. In this case we can give more precision to the general
proposition by determining the analytical character which must be
necessarily presented by the equation, in order that this property may
be verified. It is easy to see, then, that, by the modification just
explained, all the _terms_ of the first degree, whatever may be their
form, rational or irrational, entire or fractional, will become _m_
times greater; all those of the second degree, _m²_ times; those of the
third, _m³_ times, &c. Thus the terms of the same degree, however
different may be their composition, varying in the same manner, and the
terms of different degrees varying in an unequal proportion, whatever
similarity there may be in their composition, it will be necessary, to
prevent the equation from being disturbed, that all the terms which it
contains should be of the same degree. It is in this that properly
consists the ordinary theorem of _Homogeneity_, and it is from this
circumstance that the general law has derived its name, which, however,
ceases to be exactly proper for all other functions.
In order to treat this subject in its whole extent, it is important to
observe an essential condition, to which attention must be paid in
applying this property when the phenomenon expressed by the equation
presents magnitudes of different natures. Thus it may happen that the
respective units are completely independent of each other, and then the
theorem of Homogeneity will hold good, either with reference to all the
corresponding classes of quantities, or with regard to only a single one
or more of them. But it will happen on other occasions that the
different units will have fixed relations to one another, determined by
the nature of the question; then it will be necessary to pay attention
to this subordination of the units in verifying the homogeneity, which
will not exist any longer in a purely algebraic sense, and the precise
form of which will vary according to the nature of the phenomena. Thus,
for example, to fix our ideas, when, in the analytical expression of
geometrical phenomena, we are considering at once lines, areas, and
volumes, it will be necessary to observe that the three corresponding
units are necessarily so connected with each other that, according to
the subordination generally established in that respect, when the first
becomes _m_ times greater, the second becomes _m²_ times, and the third
_m³_ times. It is with such a modification that homogeneity will exist
in the equations, in which, if they are _algebraic_, we will have to
estimate the degree of each term by doubling the exponents of the
factors which correspond to areas, and tripling those of the factors
relating to volumes.
* * * * *
Such are the principal general considerations relating to the _Calculus
of Direct Functions_. We have now to pass to the philosophical
examination of the _Calculus of Indirect Functions_, the much superior
importance and extent of which claim a fuller development.