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

    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