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wholly different phenomena, through the analogies presented by the dif-
ferential expressions of their mathematical laws. In virtue of this second
property of the analysis, the entire system of an immense science, geom-
etry or mechanics, has submitted to a condensation into a small number
of analytical formulas, from which the solution of all particular prob-
lems can be deduced, by invariable rules.
This beautiful method is, however, imperfect in its logical basis. At
first, geometers were naturally more intent upon extending the discov-
ery and multiplying its applications than upon establishing the logical
foundation of its processes. It was enough for some time to be able to
produce, in answer to objections, unhoped-for solutions of the most dif-
ficult problems. It became necessary, however, to recur to the basis of
the new analysis, to establish the rigorous exactness of the processes
employed, notwithstanding their apparent breaches of the ordinary laws
of reasoning, Leibnitz himself failed to justify his conception, giving,
when urged, an answer which represented it as a mere approximative
calculus, the successive operations of which might, it is evident, admit
an augmenting amount of error. Some of his successors were satisfied
with showing that its results accorded with those obtained by ordinary
algebra, or the geometry of the ancients, reproducing by these last some
solutions which could be at first obtained only by the new method. Some,
again, demonstrated the conformity of the new conception with others;
that of Newton especially, which was unquestionably exact. This af-
82/Auguste Comte
forded a practical justification but, in a case of such unequalled impor-
tance, a logical justification is also required, a direct proof of the nec-
essary rationality of the infinitesimal method. It was Carnot who fur-
nished this at last, by showing that the method was founded on the prin-
ciple of the necessary compensation of errors. We cannot say that all the
logical scaffolding of the infinitesimal method may not have a merely
provisional existence, vicious as it is in its nature but, in the present
state of our knowledge Carnot s principle of the necessary compensa-
tion of errors is of more importance, in legitimating the analysis of
Liebnitz, than is even vet commonly supposed. His reasoning is founded
on the conception of infinitesimal quantities indefinitely decreasing, while
those from which they are derived are fixed. The infinitely small errors
introduced with the auxiliaries cannot have occasioned other than infi-
nitely small errors in all the equations; and when the relations of finite
quantities are reached, these relations must be rigorously exact, since
the only errors then possible must be finite ones, which cannot have
entered: and thus the final equations become perfect. Carnot s theory is
doubtless more subtle than solid; but it has no other radical logical vice
than that of the infinitesimal method itself, of which it is, as it seems to
me, the natural develop meet. and general explanation; so that it must be
adopted as long as that method is directly employed.
The philosophical character of the transcendental analysis has now
been sufficiently exhibited to allow of my giving only the principal idea
of the other two methods.
Newton offered his conception under several different forms in suc-
cession. That which is now most commonly adopted, at least on the
continent, was called by himself, sometimes the Method of prime and
ultimate Ratios, sometimes the Method of Limits, by which last term it
is now usually known.
Under this Method, the auxiliaries introduced are the limits of the
ratios of the simultaneous increments of the primitive quantities; or, in
other words, the final ratios of these increments, limits or final ratios
which we can easily show to have a determinate and finite value. A
special calculus, which is the equivalent of the infinitesimal calculus, is
afterwards employed, to rise from the equations between these limits to
the corresponding equations between the primitive quantities themselves.
The power of easy expression of the mathematical laws of phenom-
ena given by this analysis arises from the calculus applying, not to the
increments themselves of the proposed quantities, but to the limits of the
Positive Philosophy/83
ratios of those increments, and from our being therefore able always to
substitute for each increment any other magnitude more east to treat,
provided their final ratio is the ratio of equality or, in other words, that
the limit of their ratio is unity. It is clear, in fact, that the calculus of
limits can be in no way affected by this substitution. Starting from this
principle. We find nearly the equivalent of the facilities offered by the
analysis of Leibnitz, which are merely considered from another point of
view. Thus, curves will be regarded as the limits of a series of rectilinear
polygons, and variable motions as the limits of an aggregate of uniform
motions of continually nearer approximation, etc., etc. Such is, in sub-
stance, Newton s conception, or rather, that which Maclaurin and
d Alembert have offered as the most rational basis of the transcendental
analysis, in the endeavour to fill and arrange Newton s ideas on the
subject.
Newton had another view, however, which ought to be presented
here, because it is still the special form of the calculus of indirect func-
tions commonly adopted by English geometers; and also, on account of
its ingenious clearness in some cases and of its having furnished the
notation best adapted to this manner of regarding the transcendental
analysis. I mean the Calculus of fluxions and of fluents, founded on the
general notion of velocities.
To facilitate the conception of the fundamental idea, let us conceive
of every curve as generated by a point affected by a motion varying
according to any lank whatever. The different quantities presented by
the curve, the abscissa, the ordinate, the arc, the area, etc., will be re-
garded as simultaneously produced by successive degrees during this
motion. The velocity with which each one will have been described will
be called the fluxion of that quantity, which inversely would have been
called its fluent. Henceforth, the transcendental analysis will, according
to this conception, consist in forming directly the equations between the
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