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limits, which was the chief difficulty of the question. The task remain-
ing for modern geometers was to generalize the conception of the an-
cients, and, considering it in an ate. street manner, to reduce it to a
system of calculation, which was impossible to them.
Lagrange justly ascribes to the great geometer Fermat the first idea
in this new direction. Fermat may be regarded as having initiated the
direct formation of transcendental analysis by his method for the deter-
mination of maxima and minima, and for the finding of tangents, in
which process he introduced auxiliaries which he afterwards suppressed
as null when the equations obtained had underdone certain suitable trans-
formations. After some modifications of the ideas of Fermat in the inter-
mediate time, Leitnitz stripped the process of some complications, and
formed the analysis into a general and distinct calculus, having his own
notation: and Leibnitz is thus the creator of transcendental analysis, as
we employ it now. This pre-eminent discovery was so ripe, as all great
conceptions are at the hour of their advent, that Newton had at the same
time, or rather earlier, discovered a method exactly equivalent, regard-
ing the analysis from a different point of view, much more logical in
itself, but less adapted than that of Leitnitz to give all practicable extent
and facility to the fundamental method. Lagrange afterwards, discard-
ing the heterogeneous considerations which had guided Leibnitz and
Newton, reduced the analysis to a purely algebraic system, which only
wants more aptitude for application.
We will notice the three methods in their order. The method of
Leibnitz consists in introducing into the calculus, in order to facilitate
the establishment of equations, the infinitely small elements or differen-
tials which are supposed to constitute the quantities whose relations we
are seeking. There are relations between these differentials which are
simpler and more discoverable than those of the primitive quantities;
and by these we may afterwards (through a special calculus employed
to eliminate these auxiliary infinitesimals) recur to the equations sought,
which it would usually have been impossible to obtain directly. This
indirect analysis may have various decrees of indirectness for, when
there is too much difficulty in forming the equation between the differ-
entials of the magnitudes under notice, a second application of the method
is required, the differentials being now treated as new primitive quanti-
ties, and a relation being sought between their infinitely small elements,
80/Auguste Comte
or second differentials, and so on; the same transformation being re-
peated any number of times, provided the whole number of auxiliaries
be finally eliminated.
It may be asked by novices in these studies, how these auxiliary
quantities can be of use while they are of the same species with the
magnitudes to be treated, seeing that the greater or less value of any
quantity cannot affect any inquiry which has nothing to do with value at
all. The explanation is this. We must begin by distinguishing the differ-
ent orders of infinitely small quantities, obtaining a precise idea of this
by considering them as being, either the successive powers of the same
primitive infinitely small quantity, or as being quantities which may be
regarded as having finite ratios with these powers; so that, for instance
the second or third or other differentials of the same variable are classed
as infinitely small quantities of the second, third or other order, because
it is easy to exhibit in them finite multiples of the second, third, or other
powers of a certain first differential. These preliminary ideas being laid
down the spirit of the infinitesimal analysis consists in constantly ne-
glecting the infinitely small quantities in comparison with finite quanti-
ties; and generally, the infinitely small quantities of any order whatever
in comparison with all those of an inferior order. We see at once how
such a prover must facilitate the formation of equations between the
differentials of quantities, since we can substitute for these differentials
such other elements as we may choose, and as will be more simple to
treat, only observing the condition that the new elements shall differ
from the preceding only by quantities infinitely small in relation to them.
It is thus that it becomes possible in geometry to treat curve: lines as
composed of an infinity of rectilinear elements, and curved surfaces as
formed of plane elements; and, in mechanics, varied motions as an infi-
nite series of uniform motions, succeeding each other at infinitely small
intervals of time. Such a mere hint as this of the varied application of
this method may give some idea of the vast scope of the conception of
transcendental analysis, as formed by Leibnitz. It is, beyond all ques-
tion, the loftiest idea ever yet attained by the human mind.
It is clear that this conception was necessary to complete the basis
of mathematical science, by enabling, us to estate fish, in a broad and
practical manner, the relation of the concrete to the abstract. In this
respect, we must regard it as the necessary complement of the great
fundamental idea of Descartes on the general analytical representation
of natural phenomena; an idea which could not be duly estimated or put
Positive Philosophy/81
to use till after the formation of the infinitesimal analysis.
This analysis has another property, besides that of facilitating the
study of the mathematical laws of all phenomena, and perhaps not less
important than that. The differential formulas exhibit an extreme gener-
ality, expressing in a single equation each determinate phenomenon,
however varied may be the subjects to which it belongs. Thus, one such
equation gives the tangents of all curves, another their rectifications, a
third their quadratures; and, in the same way, one invariable formula
expresses the mathematical law of all variable motion; and one single
equation represents the distribution of heat in any body, and for any
case. This remarkable generality is the basis of the loftiest views of the
geometers. Thus this analysis has not only furnished a general method
for forming equations indirectly which could not have been directly dis-
covered, but it teas introduced a new order of more natural laws for our
use in the mathematical study of natural phenomena, enabling us to rise [ Pobierz całość w formacie PDF ]
zanotowane.pl doc.pisz.pl pdf.pisz.pl szkicerysunki.xlx.pl
limits, which was the chief difficulty of the question. The task remain-
ing for modern geometers was to generalize the conception of the an-
cients, and, considering it in an ate. street manner, to reduce it to a
system of calculation, which was impossible to them.
Lagrange justly ascribes to the great geometer Fermat the first idea
in this new direction. Fermat may be regarded as having initiated the
direct formation of transcendental analysis by his method for the deter-
mination of maxima and minima, and for the finding of tangents, in
which process he introduced auxiliaries which he afterwards suppressed
as null when the equations obtained had underdone certain suitable trans-
formations. After some modifications of the ideas of Fermat in the inter-
mediate time, Leitnitz stripped the process of some complications, and
formed the analysis into a general and distinct calculus, having his own
notation: and Leibnitz is thus the creator of transcendental analysis, as
we employ it now. This pre-eminent discovery was so ripe, as all great
conceptions are at the hour of their advent, that Newton had at the same
time, or rather earlier, discovered a method exactly equivalent, regard-
ing the analysis from a different point of view, much more logical in
itself, but less adapted than that of Leitnitz to give all practicable extent
and facility to the fundamental method. Lagrange afterwards, discard-
ing the heterogeneous considerations which had guided Leibnitz and
Newton, reduced the analysis to a purely algebraic system, which only
wants more aptitude for application.
We will notice the three methods in their order. The method of
Leibnitz consists in introducing into the calculus, in order to facilitate
the establishment of equations, the infinitely small elements or differen-
tials which are supposed to constitute the quantities whose relations we
are seeking. There are relations between these differentials which are
simpler and more discoverable than those of the primitive quantities;
and by these we may afterwards (through a special calculus employed
to eliminate these auxiliary infinitesimals) recur to the equations sought,
which it would usually have been impossible to obtain directly. This
indirect analysis may have various decrees of indirectness for, when
there is too much difficulty in forming the equation between the differ-
entials of the magnitudes under notice, a second application of the method
is required, the differentials being now treated as new primitive quanti-
ties, and a relation being sought between their infinitely small elements,
80/Auguste Comte
or second differentials, and so on; the same transformation being re-
peated any number of times, provided the whole number of auxiliaries
be finally eliminated.
It may be asked by novices in these studies, how these auxiliary
quantities can be of use while they are of the same species with the
magnitudes to be treated, seeing that the greater or less value of any
quantity cannot affect any inquiry which has nothing to do with value at
all. The explanation is this. We must begin by distinguishing the differ-
ent orders of infinitely small quantities, obtaining a precise idea of this
by considering them as being, either the successive powers of the same
primitive infinitely small quantity, or as being quantities which may be
regarded as having finite ratios with these powers; so that, for instance
the second or third or other differentials of the same variable are classed
as infinitely small quantities of the second, third or other order, because
it is easy to exhibit in them finite multiples of the second, third, or other
powers of a certain first differential. These preliminary ideas being laid
down the spirit of the infinitesimal analysis consists in constantly ne-
glecting the infinitely small quantities in comparison with finite quanti-
ties; and generally, the infinitely small quantities of any order whatever
in comparison with all those of an inferior order. We see at once how
such a prover must facilitate the formation of equations between the
differentials of quantities, since we can substitute for these differentials
such other elements as we may choose, and as will be more simple to
treat, only observing the condition that the new elements shall differ
from the preceding only by quantities infinitely small in relation to them.
It is thus that it becomes possible in geometry to treat curve: lines as
composed of an infinity of rectilinear elements, and curved surfaces as
formed of plane elements; and, in mechanics, varied motions as an infi-
nite series of uniform motions, succeeding each other at infinitely small
intervals of time. Such a mere hint as this of the varied application of
this method may give some idea of the vast scope of the conception of
transcendental analysis, as formed by Leibnitz. It is, beyond all ques-
tion, the loftiest idea ever yet attained by the human mind.
It is clear that this conception was necessary to complete the basis
of mathematical science, by enabling, us to estate fish, in a broad and
practical manner, the relation of the concrete to the abstract. In this
respect, we must regard it as the necessary complement of the great
fundamental idea of Descartes on the general analytical representation
of natural phenomena; an idea which could not be duly estimated or put
Positive Philosophy/81
to use till after the formation of the infinitesimal analysis.
This analysis has another property, besides that of facilitating the
study of the mathematical laws of all phenomena, and perhaps not less
important than that. The differential formulas exhibit an extreme gener-
ality, expressing in a single equation each determinate phenomenon,
however varied may be the subjects to which it belongs. Thus, one such
equation gives the tangents of all curves, another their rectifications, a
third their quadratures; and, in the same way, one invariable formula
expresses the mathematical law of all variable motion; and one single
equation represents the distribution of heat in any body, and for any
case. This remarkable generality is the basis of the loftiest views of the
geometers. Thus this analysis has not only furnished a general method
for forming equations indirectly which could not have been directly dis-
covered, but it teas introduced a new order of more natural laws for our
use in the mathematical study of natural phenomena, enabling us to rise [ Pobierz całość w formacie PDF ]