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pacity of the transcendental analysis for the establishment of equations
in the most difficult problems, by considering a class of equations still
more indirect than differential equations properly so called. It is still too
near its origin, and its applications have been too few, to admit of its
being understood by a purely abstract account of its theory; and it is
therefore necessary to indicate briefly the special nature of the problems
which have given rise to this hyper-transcendental analysis.
These problems are those which were long known by the name of
Isoperimetrical Problems, a name which is truly applicable to only a
very small number of them. They consist in the investigation of the
maxima and minima of certain indeterminate integral formulas which
express the analytical law of such or such a geometrical or mechanical
phenomenon, considered independently of any particular subject.
In the ordinary theory of maxima and minima, we seek, with regard
to a given function of one or more variables, what particularvalues must
be assigned to these variables, in order that the corresponding value of
the proposed function may be a maximum or a minimum with respect to
those values which immediately precede and follow it: that is, we in-
quire, properly speaking, at what instant the function ceases to increase
in order to begin to decrease, or the reverse. The differential calculus
fully suffices, as we know, for the general resolution of this class of
questions, by showing that the values of the different variables which
suit either the maximum or minimum must always render null the differ-
ent derivatives of the first order of the given function, taken separately
with relation to each independent variable, and by indicating moreover
a character suitable for distinguishing the maximum from the minimum,
which consists, in the case of a function of a single variable, for ex-
ample, in the derived function of the second order taking a negative
value for the maximum and a positive for the minimum. Such are the
fundamental conditions belonging to the majority of cases; and where
modifications take place, they are equally subject to invariable, though
more complicated abstract rules.
The construction of this general theory haying destroyed the chief
interests of geometers in this kind of questions they rose almost immedi-
ately to the consideration of a new order of problems, at once more
important and more difficult, those of isoperimeters. It was then no
longer the values of the variables proper to the maximum or the mini-
Positive Philosophy/99
mum of a given function that had to be determined. It was the form of
the function itself that had to be discovered, according to the condition
of the maximum or minimum of a certain definite integral, merely indi-
cated, which depended on that function. We cannot here follow the his-
tory of these problems, the oldest of which is that of the solid of least
resistance, treated by Newton in the second book of the Principia, in
which he determines what must be the meridian curve of a solid of revo-
lution in order that the resistance experienced by that body in the direc-
tion of its axis may be the least possible. Mechanics first furnished this
new class of problems; but it was from geometry that the subjects of the
principal investigations were afterwards derived. They were varied and
complicated almost infinitely by the labours of the best geometers, when
Lagrange reduced their solution to an abstract and entirely general
method, the discovery of which has checked the eagerness of geometers
about such an order of researches.
It is evident that these problems, considered analytically, consist in
determining what ought to be the form of a certain unknown function of
one or more variables, in order that such or such an integral, dependent
on that function, may have, within assigned limits, a value which may
be a maximum or a minimum, with regard to all those which it would
take if the required function had any other form whatever. In treating
these problems, the predecessors of Lagrange proposed, in substance,
to reduce them to the ordinary theory of maxima and minima. But they
proceeded by applying special simple artifices to each case, not reduc-
ible to certain rules; so that every new question reproduced analogous
difficulties, without the solutions previously obtained being of any es-
sential aid. The part common to all questions of this class had not been
discovered; and no abstract and general treatment was therefore pro-
vided. In his endeavours to bring all isoperimetrical problems to depend
on a common analysis Lagrange was led to the conception of a new kind
of differentiation, and to these new Differentials he gave the name of
Variations. They consist of the infinitely small increments which the
integrals receive, not in virtue of analogous increments on the part of
the corresponding variables, as in the common transcendental analysis,
but by supposing that the form of the function placed under the sign of
integration undergoes an infinitely small change. This abstract concep-
tion once formed, Lagrange was able to reduce with ease, and in the
most general manner, all the problems of isoperimeters to the simple
common theory of maxima and minima.
100/Auguste Comte
Important as is this great and happy transformation, and though the
Method of Variations had at first no other object than the rational and
general resolution of isoperimetrical problems, we should form a very
inadequate estimate of this beautiful analysis if we supposed it restricted
to this application. In fact, the abstract conception of two distinct na-
tures of differentiation is evidently applicable, not only to the cases for
which it was created but for all which present, for any reason whatever,
two different ways of making the same magnitudes vary. Lagrange him-
self made an immense and all-important application of his Calculus of
Variations, in his Analytical Mechanics, by employing it to distin-
guish the two sorts of changes, naturally presented by questions of ra-
tional Mechanics for the different points we have to consider, according
as we compare the successive positions occupied, in virtue of its mo-
tion, by the same point of each body in two consecutive instants, or as
we pass from one point of the body to another in the same instant. One
of these comparisons produces the common differentials; the other oc-
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