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rived from the primary. It is true, in the majority of important cases, its
use requires to be preceded and prepared for by that of the calculus of
indirect functions, by which the establishment of equations is facili-
tated: but though algebra then takes the second place, it is not the less a
necessary agent in the solution of the question; so that the Calculus of
direct functions must continue to be, by its nature, the basis of math-
ematical analysis. We must now, then, notice the rational composition
of this calculus, and the degree of development it has attained.
Its object being the resolution of equations (that is, the discovery of
the mode of formation of unknown quantities by the known, according
to the equations which exist between them), it presents as many parts as
we can imagine distinct classes of equations; and its extent is therefore
rigorously indefinite, because the number of analytical functions sus-
ceptible of entering into equations is illimitable, though, as we have
seen, composed 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. Accordingly, analysts first divide equations with one or more
variables into two principal classes, according as they contain functions
of only the first three of the ten couples, or as they include also either
exponential or circular functions. Though the names of algebraic and
transcendental functions given to these principal groups are inapt, the
division between the corresponding equations is real enough, insofar as
that the resolution of equations containing the transcendental functions
is more difficult than that of algebraic equations. Hence the study of the
first is extremely imperfect, and our analytical methods relate almost
exclusively to the elaboration of the second
Our business now is with these Algebraic equations only. In the
first place, we must observe that, though they may often contain irratio-
nal functions of the unknown quantities, as well as rational functions,
the first case can always be brought under the second, by transforma-
Positive Philosophy/75
tions more or less easy so that it is only with the latter that analysts have
had to occupy themselves, to resolve all the algebraic equations. As to
their classification, the early method of classing them according to the
number of their terms has been retained only for equations with two
terms, which are, in fact, susceptible of a resolution proper to them-
selves. The classification by their degrees, long universally established,
is eminently natural; for this distinction rigorously determines the greater
or less difficulty of their resolution. The gradation can be independently,
as well as practically exhibited: for the most general equation of each
degree necessarily comprehends all those of the different inferior de-
grees, as must also the formula which determines the unknown quantity:
and therefore, however slight we may, a priori, suppose the difficulty to
be of the degree under notice, it must offer more and more obstacles, in
proportion to the rank of the degree, because it is complicated in the
execution with those of all the preceding degrees.
This increase of difficulty is so great, that the resolution of alge-
braic equations is as yet known to us only in the four first degrees. In
this respect, algebra has advanced but little since the labours of Descartes
and the Italian analysts of the sixteenth century; though there has prob-
ably not been a single geometer for two centuries past who has not
striven to advance the resolution of equations. The general equation of
the fifth degree has itself, thus far, resisted all attempts. The formula of
the fourth degree is so difficult as to be almost inapplicable; and ana-
lysts, while by no means despairing of the resolution of equations of the
fifth, and even higher degrees, being obtained, have tacitly agreed to
give up such researches.
The only question of this kind which would be of eminent impor-
tance, at least in its logical relations, would be the general of algebraic
equations of any degree whatever. But the more we ponder this subject,
the more we are led to suppose. with Lagrange, that it exceeds the scope
of our understandings. Even if the requisite formula could be obtained,
it could not be usefully applied unless we could simplify it, without
impairing its generality, by the introduction of a new class of analytical
elements, of which we have as yet no idea. And, besides, if we had
obtained the resolution of algebraic equations of any degree whatever,
we should still have treated only a very small part of algebra, properly
so called; that is, of the calculus of direct functions, comprehending the
resolution of all the equations that can be formed bv the analytical func-
tions known to us at this day. Again, we must remember that by a law of
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our nature, we shall always remain below the difficulty of science, our
means of conceiving of new questions being always more powerful than
our resources for resolving them; in other words, the human mind being
more apt at imagining than at reasoning. Thus, if we had resolved all the
analytical equations now known, and if, to do this, we had found new
analytical elements, these again would introduce classes of equations of
which we now know nothing: and so, however great might be the in-
crease of our knowledge, the imperfection of our algebraic science would
be perpetually reproduced.
The methods that we have are, the complete resolution of the equa-
tions of the first four degrees; of any binomial equations; of certain
special equations of the superior degrees; and of a very small number of
exponential, logarithmic, and circular equations. These elements are very
limited; but geometers have succeeded in treating with them a great
number of important questions in an admirable manner. The improve-
ments introduced within a century into mathematical analysis have con-
tributed more to render the little knowledge that we have immeasurably
useful, than to increase it.
To fill up the vast gap in the resolution of algebraic equations of the
higher degrees, analysts have had recourse to a new order of questions,
to what they call the numerical resolution of equations. Not being able
to obtain the real algebraic formula, they have sought to determine at
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