of Brook Taylor and Colin Maclaurin were absorbed into the rapidly growing continental analysis, and the more precise conceptions reached through a critical scrutiny of the true nature of Newton’s fluxions and moments stimulated a like scrutiny of the basis of the method of differentials.
30. This method had met with opposition from the first.
Christiaan Huygens, whose opinion carried more weight than
that of any other scientific man of the day, declared
that the employment of differentials was unnecessary,
and that Leibnitz’s second differential was meaningless
Opposition to
the calculus.
(1691). A Dutch physician named Bernhard Nieuwentijt
attacked the method on account of the use of quantities
which are at one stage of the process treated as somethings and
at a later stage as nothings, and he was especially severe in
commenting upon the second and higher differentials (1694, 1695).
Other attacks were made by Michel Rolle (1701), but they
were directed rather against matters of detail than against the
general principles. The fact is that, although Leibnitz in his
answers to Nieuwentijt (1695), and to Rolle (1702), indicated
that the processes of the calculus could be justified by the
methods of the ancient geometry, he never expressed himself
very clearly on the subject of differentials, and he conveyed,
probably without intending it, the impression that the calculus
leads to correct results by compensation of errors. In England
the method of fluxions had to face similar attacks. George
Berkeley, bishop and philosopher, wrote in 1734 a tract entitled
The Analyst; or a Discourse addressed to an Infidel Mathematician,
in which he proposed to destroy the presumption that the
The “Analyst” controversy.
opinions of mathematicians in matters of faith are
likely to be more trustworthy than those of divines,
by contending that in the much vaunted fluxional
calculus there are mysteries which are accepted
unquestioningly by the mathematicians, but are incapable of
logical demonstration. Berkeley’s criticism was levelled against
all infinitesimals, that is to say, all quantities vaguely conceived
as in some intermediate state between nullity and finiteness,
as he took Newton’s moments to be conceived. The tract
occasioned a controversy which had the important consequence
of making it plain that all arguments about infinitesimals must
be given up, and the calculus must be founded on the method of
limits. During the controversy Benjamin Robins gave an
exceedingly clear explanation of Newton’s theories of fluxions
and of prime and ultimate ratios regarded as theories of limits.
In this explanation he pointed out that Newton’s moment
(Leibnitz’s “differential”) is to be regarded as so much of the
actual difference between two neighbouring values of a variable
as is needful for the formation of the fluxion (or differential
coefficient) (see G. A. Gibson, “The Analyst Controversy,”
Proc. Math. Soc., Edinburgh, xvii., 1899). Colin Maclaurin
published in 1742 a Treatise of Fluxions, in which he reduced
the whole theory to a theory of limits, and demonstrated it by
the method of Archimedes. This notion was gradually transferred
to the continental mathematicians. Leonhard Euler
in his Institutiones Calculi differentialis (1755) was reduced to the
position of one who asserts that all differentials are zero, but,
as the product of zero and any finite quantity is zero, the ratio
of two zeros can be a finite quantity which it is the business
of the calculus to determine. Jean le Rond d’Alembert in the
Encyclopédie méthodique (1755, 2nd ed. 1784) declared that
differentials were unnecessary, and that Leibnitz’s calculus was
a calculus of mutually compensating errors, while Newton’s
method was entirely rigorous. D’Alembert’s opinion of Leibnitz’s
calculus was expressed also by Lazare N. M. Carnot in his
Réflexions sur la métaphysique du calcul infinitésimal (1799)
and by Joseph Louis de la Grange (generally called Lagrange)
in writings from 1760 onwards. Lagrange proposed in his
Théorie des fonctions analytiques (1797) to found the whole of the
calculus on the theory of series. It was not until 1823 that a
treatise on the differential calculus founded upon the method
of limits was published. The treatise was the Résumé des leçons
Cauchy’s method
of limits.
. . . sur le calcul infinitésimal of Augustin Louis Cauchy.
Since that time it has been understood that the use of the
phrase “infinitely small” in any mathematical argument
is a figurative mode of expression pointing to a
limiting process. In the opinion of many eminent
mathematicians such modes of expression are
confusing to students, but in treatises on the
calculus the traditional modes of expression are still largely
adopted.
31. Defective modes of expression did not hinder constructive
work. It was the great merit of Leibnitz’s symbolism that
a mathematician who used it knew what was to be
done in order to formulate any problem analytically,
even though he might not be absolutely clear as to the
Arithmetical basis
of modern analysis.
proper interpretation of the symbols, or able to render
a satisfactory account of them. While new and varied
results were promptly obtained by using them, a long time elapsed
before the theory of them was placed on a sound basis. Even
after Cauchy had formulated his theory much remained to be
done, both in the rapidly growing department of complex
variables, and in the regions opened up by the theory of expansions
in trigonometric series. In both directions it was seen
that rigorous demonstration demanded greater precision in
regard to fundamental notions, and the requirement of precision
led to a gradual shifting of the basis of analysis from geometrical
intuition to arithmetical law. A sketch of the outcome of this
movement—the “arithmetization of analysis,” as it has been
called—will be found in Function. Its general tendency has
been to show that many theories and processes, at first accepted
as of general validity, are liable to exceptions, and much of the
work of the analysts of the latter half of the 19th century was
directed to discovering the most general conditions in which
particular processes, frequently but not universally applicable,
can be used without scruple.
III. Outlines of the Infinitesimal Calculus.
32. The general notions of functionality, limits and continuity are explained in the article Function. Illustrations of the more immediate ways in which these notions present themselves in the development of the differential and integral calculus will be useful in what follows.
33. Let y be given as a function of x, or, more generally, let x
and y be given as functions of a variable t. The first of these cases
is included in the second by putting x = t. If certain
conditions are satisfied the aggregate of the points determined
by the functional relations form a curve. The
Geometrical limits.
first condition is that the aggregate of the values of t to
which values of x and y correspond must be continuous, or, in other
words, that these values must consist of all real numbers, or of all
those real numbers which lie between assigned extreme numbers.
When this condition is satisfied the points are “ordered,” and their
order is determined by the order of the numbers t, supposed to be
arranged in order of increasing or decreasing magnitude; also
Fig. 8.
there are two senses of description of the curve, according as t is
taken to increase or to diminish. The second condition is that
the aggregate of the points which are determined by the functional
relations must be “continuous.” This condition means that, if
any point P determined by a value of t is taken, and any distance δ,
however small, is chosen, it is possible to find two points Q, Q′ of the
aggregate which are such that (i.) P is between Q and Q′, (ii.) if
R, R′ are any points between Q and Q′ the distance RR′ is less
than δ. The meaning of the word “between” in this statement
is fixed by the ordering of the points. Sometimes additional conditions
are imposed upon the functional relations before they are
regarded as defining a curve. An aggregate of points which satisfies
the two conditions stated
above is sometimes called a
“Jordan curve.” It by no
means follows that every
curve of this kind has a tangent.
In order that the curve
may have a tangent
at P it is necessaryTangents.
that, if any angle α, however
small, is specified, a distance δ
can be found such that when
P is between Q and Q′, and
PQ and PQ′ are less than δ,
the angle RPR′ is less than
α for all pairs of points R, R′ which are between P and Q, or
between P and Q′ (fig. 8). When this condition is satisfied y is a
function of x which has a differential coefficient. The only way of
