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LEIBNIZ, GOTTFRIED WILHELM (b. Leipzig,
Germany, 1 July 1646; d. Hannover, Germany,
14 November 1716), mathematics, philosophy, metaphysics.

LEIBNIZ: Mathematics

assembled was based not on the largely inaccessible
originals but, rather, on copies in which crucial
passages were abbreviated. As a result, it was possible
for the reader to gain the impression that Newton
possessed priority in having obtained decisive results
in the field of infinitesimals (method of tangents,
power series in the handling of quadratures, and
inverse tangent problems) and that Leibniz was
guilty of plagiarism on the basis of what he had taken
from Newton. Fatio pronounced this reproach in the
sharpest terms in his Lineae brevissimi descensus
investigatio (1699).

Leibniz replied in 1700 with a vigorous defense of
his position, in which he stressed that he had obtained
only results, not methods, from Newton and that he
had already published the fundamental concepts of
the differential calculus in 1684, three years before
the appearance of the material that Newton referred
to in a similar form in his Principia (1687). On this
occasion he also described his own procedure with
reference to de Moivre's theorem (1698) on series
inversion through the use of undetermined coefficients.
Leibniz made his procedure more general and easier
to grasp by the introduction of numerical coefficients
(in the sense of indices). The attack subsided because
Fatio, who was excitable, oversensitive, and given to
a coarse manner of expression, turned away from
science and became a fervent adherent of an aggressive
religious sect. He was eventually pilloried.

Wallis' insinuations were repeated by G. Cheyne
in his Methodus (1703) and temperately yet firmly
rebutted in Leibniz' review of 1703. Cheyne's discussion
of special quadratures was probably what led
Newton to publish the Quadratura curvarum (manuscript
of 1676) and the Enumeratio linearum tertii
ordinis (studies beginning in 1667-1668) as appendices
to the Optics (1704). Newton viewed certain passages
of Leibniz' review of the Optics (1705) as abusive
attacks and gave additional material to John Keill,
who, in a paper published in 1710, publicly accused
Leibniz of plagiarism. Leibniz' protests (1711) led to
the establishment of a commission of the Royal
Society, which decided against him (1712) on the
basis of the letters printed by Wallis and further
earlier writings produced by Newton. The commission
published the evidence, together with an analysis that
had been published in 1711, in a Commercium epistolicum
(1713 edition).

The verdict reached by these biased investigators,
who heard no testimony from Leibniz and only
superficially examined the available data, was accepted
without question for some 140 years and was
influential into the first half of the twentieth century.
In the light of the much more extensive material now
available, it is recognized as wrong. It can be understood
only in the nationalistic context in which the
controversy took place. The continuation of the
quarrel was an embarrassment to both parties; and,
since it yielded nothing new scientifically, it is unimportant
for an understanding of Leibniz' mathematics.
The intended rebuttal did not materialize,
and Leibniz' interesting, but fragmentary, account of
how he arrived at his discovery (Historia et origo
calculi differentialis [1714]) was not published until
the nineteenth century.

The hints concerning mathematical topics in
Leibniz' correspondence are especially fascinating.
When writing to those experienced in mathematics,
whom he viewed as competitors, he expressed himself
very cautiously, yet with such extraordinary cleverness
that his words imply far more than is apparent from
an examination of the notes and jottings preserved
in his papers. For instance, his remarks on the
solvability of higher equations are actually an anticipation
of Galois's theory. Frequently, material of
general validity is illustrated only by simple examples,
as is the case with the schematic solution of systems of
linear equations by means of number couples (double
indices) in quadratic arrangement, which corresponds
to the determinant form (1693).

In several places the metaphysical background is
very much in evidence, as in the working out of
binary numeration, which Leibniz connected with
the creation of the world (indicated by 1) from
nothingness (indicated by 0). The same is true of his
interpretation of the imaginary number as an intermediate
entity between Being and Not-Being (1702).
His hope of being able to make a statement about
the transcendence of π by employing the dyadic
presentation (1701) was fruitless yet interesting, for
transcendental numbers can be constructed out of
infinite dual fractions possessing regular gaps (an
example in the decimal system was given by Goldbach
in 1729). The attempts to present ?r(n=1) 1/n (reference in
1682, recognized as false in 1696) and
??(n=1) 1/n2
(1696)
in closed form were unsuccessful, and the claimed
rectification of an arc of the equilateral hyperbola
through the quadrature of the hyperbola (1676) was
based on an error in computation. Against these
failures we may set the importance of the recognition
of the correspondence between the multinomial
theorem (1676) and the continuous differentiation
and integration with fractional index that emerged
from this observation. During this period (about
1696) Leibniz also achieved the general representation