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ing the connection of the different bodies of the system, in virtue of
which their mutual reactions will necessarily affect the motions which
each would take if alone; and we can have no a priori knowledge of
what the alterations will be. In the case of the pendulum, for instance,
the particles nearest the point of suspension, and those furthest from it,
must react on each other by their connection, the one moving faster
and the other slower than if they had been free; and no established dy-
namic principle exists revealing the law which determines these reac-
tions. Geometers naturally began by laying down a principle for each
particular case; and many were the principles thus offered, which turned
out to be only remarkable theorems furnished simultaneously by funda-
mental dynamic equations. Lagrange has given us, in his Analytical
Mechanics, the general history of this series of labours: and very inter-
esting it is, as a study of the progressive march of the human intellect.
This method of proceeding continued till the time of D Alembert, who
put an end to all these isolated researches by seeing how to compute the
reactions of the bodies of a system in virtue of their connection, and
establishing the fundamental equations of the motion of any system. By
the aid of the great principle which bears his name, he made questions
of motion merge in simple questions of equilibrium. The principle is
simply this. In the case supposed, the natural motion clearly divides
itself into two, the one which subsists and the one which has been
destroyed. By D Alembert s view, all these last, or, in other words, all
the motions that have been lost or gained by the different bodies of the
system by their reaction, necessarily balance each other, under the con-
ditions of the connection which characterizes the proposed system. James
142/Auguste Comte
Bernouilli saw this with regard to the particular case of the pendulum;
and he was led by it to form an equation adapted to determine the centre
of oscillation of the most simple system of weight. But he extended the
resource no further; and what he did detracts nothing from the credit of
D Alembert s conception, the excellence of which consists in its entire
generality.
In D Alembert s hands the principle seemed to have a purely logical
character. But its germ may be recognized in the second law of motion,
established by Newton, under the name of the equality of reaction and
action. They are in fact, the same, with regard to two bodies only acting
upon each other in the line which connects them. The one is the greatest
possible generalization of the other; and this way of regarding it brings
out its true nature, by giving it the physical character which D Alembert
did not impress upon it. Henceforth therefore we recognize in it the
second law of motion, extended to any number of bodies connected in
any manner.
We see how every dynamical question is thus convertible into one of
Statics, by forming, in each case, equations of equilibrium between the
destroyed motions. But then comes the difficulty of making out what the
destroyed motions are. In endeavouring to get rid of the embarrassing
consideration of the quantities of motion lost or gained Euler, above
others, has supplied us with the method most suitable for use, that of
attributing to each body a quantity of motion equal and contrary to that
which it exhibits, it being evident that if such equal and contrary motion
could be imposed upon it, equilibrium would be the result. This method
contemplates only the primitive and the actual motions which are the
true elements of the dynamic problem, the given find the unknown;
and it is under this method that D Alembert s principle is habitually
conceived of. Questions of motion being thus reduced to questions of
equilibrium, the next step is to combine D Alembert s principle with
that of virtual velocities. This is the combination proposed by Lagrange,
and developed in his Analytical Mechanics, which has carried up the
science of abstract Mechanics to the highest degree of logical perfec-
tion, that is, to a rigorous unity. All questions that it can comprehend
are brought under a single principle, through which the solution of any
problem whatever offers only analytical difficulties.
D Alembert immediately applied his principle to the case of flu-
ids liquid and gaseous, which evidently admit of its use as well as
solids, their peculiar conditions being considered. The result was our
Positive Philosophy/143
obtaining general equations of the motion of fluids, wholly unknown
before. The principal of virtual velocities rendered this perfectly easy,
and again left nothing to be desired, in regard to concrete consider-
ations, and presented none but analytical difficulties. We must admit
however that our actual knowledge obtained under this theory is ex-
tremely imperfect, owing to insurmountable difficulties in the integra-
tions required. If it was so in questions of pure Statics, much more must
it be so in the more complex dynamical questions. The problem of the
flow of a gravitating liquid through a given orifice, simple as it appears,
has never yet been resolved. To simplify as far as they could, geometers
have had recourse to Daniel Bernouilli s hypothesis of the parallelism
of sections, which admits of our considering motion in regard to hori-
zontal laminae instead of particle by particle. But this method of consid-
ering each horizontal laming of a liquid as moving altogether, and talk-
ing the place of the following, is evidently contrary to the fact in almost
all cases. The lateral motions are wholly abstracted, and their sensible
existence imposes on us the necessity of studying the motion of each
particle. We must then consider the science of hydrodynamics as being
still in its infancy, even with regard to liquids, and much more with
retard to gases. Yet, as the fundamental equations of the motions of
fluids are irreversibly established, it is clear that what remains to be
accomplished is in the direction of mathematical analysis alone.
Such is the Method of Rational Mechanics. As for the great theo-
retical results of the science, the principal general properties of equi-
librium and motion thus far discovered, they were at first, taken for
real principles, each being destined to furnish the solution of a certain
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