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

    of partial differentiation; the reduction of the differential equation

a00 + a10x + (a01 + a11x)y' = 0

by means of the transformation

x = p11u + p12v, y = p21u + P22v;

the solution of

y' + p(x)y + q(x) = 0

and

(y')2 + p(x)y' + q(x) = 0

through series with coefficients in number couples; and the reduction of the equation, which had originated with Jakob Bernoulli,

y' = p(x)y + q(x)yn

to

y' = P(x)y + Q(x).

Leibniz' inventive powers and productivity in mathematics did not begin to slow until around 1700. The integration of rational functions (1702-1703) was, to be sure, an important accomplishment; but the subject was not completely explored, since Leibniz supposed (Johann Bernoulli to the contrary) that there existed other imaginary units besides ?-1 (for example, 4?-1) which could not be represented by ordinary complex numbers. The subsequent study, which was not published at the time, on the integration of special classes of irrational functions also remains of great interest. The discussion with Johann Bernoulli on the determination of arclike algebraic curves in the plane (1704-1706) resulted in both a consideration of relative motions in the plane and an interesting geometric construction of the arclike curve equivalent to a given curve. This construction is related to the optical essay of 1689 and to the theory of envelopes of 1692-1694, but it cannot readily be grasped in terms of a formula (1706). The remarks (1712) on the logarithms of negative numbers and on the representation of ?(-1)n by 1/2 can no longer be considered satisfactory. The description of the calculating machine (1710) indicates its importance but does not give the important details; the first machines of practical application were constructed by P. M. Hahn in 1774 on the basis of Leibniz' ideas.

Leibniz accorded a great importance to mathematics because of its broad interest and numerous applications. The extent of his concern with mathematics is evident from the countless remarks and notes in his posthumous papers, only small portions of which have been accessible in print until now, as well as from the exceedingly challenging and suggestive influential comments expressed brilliantly in the letters and in the works published in his own lifetime.

Leibniz' power lay primarily in his great ability to distinguish the essential elements in the results of others, which were often rambling and presented in a manner that was difficult to understand. He put them in a new form, and by setting them in a larger context made them into a harmoniously balanced and comprehensive whole. This was possible only because in his reading Leibniz was prepared, despite his impatience, to immerse himself enthusiastically and selflessly in the thought of others. He was concerned with formulating authoritative ideas clearly and connecting them, as he did in so exemplary a fashion in the mathematics of infinitesimals. Interesting details were important—they occur often in his notes—but even more important were inner relationships and their comprehension, as this term is employed in the history of thought. He undertook the work required by such an approach for no other purpose than the exploration of the conditions under which new ideas emerge, stimulate each other, and are joined in a unified thought structure.

JOSEPH E. HOFMANN

BIBLIOGRAPHY

I. ORIGINAL WORKS.

The following volumes of the Sämtliche Schriften und Briefe, edited by the Deutsche Akademie der Wissenschaften in Berlin, have been published: Series I (Allgemeiner politischer und historischer Briefwechsel), vol. 1: 1668-1676 (1923; Hildesheim, 1970); vol. 2: 1676-1679 (1927; Hildesheim, 1970); vol. 3: 1680-1683 (1938; Hildesheim, 1970); vol. 4: 1684-1687 (1950); vol. 5: 1687-1690 (1954; Hildesheim, 1970); vol. 6: 1690-1691 (1957; Hildesheim, 1970); vol. 7: 1691-1692 (1964); vol. 8: 1692 (1970); Series II (Philosophischer Briefwechsel), vol. 1: 1663-1685 (1926); Series IV (Politische Schriften), vol. 1: 1667-1676 (1931); vol. 2: 1677-1687 (1963); Series VI (Philosophische Schriften), vol. 1: 1663-1672 (1930); vol. 2: 1663-1672 (1966); vol. 6: Nouveaux essais (1962).

Until publication of the Sämtliche Schriften und Briefe is completed, it is necessary to use earlier editions and recent partial editions. The most important of these editions are the following (the larger editions are cited first): L. Dutens, ed., G. W. Leibnitii opera omnia, 6 vols. (Geneva, 1768); J. E. Erdmann, ed., G. W. Leibniz. Opera philosophica quae extant latina gallica germanica omnia (Berlin, 1840; Aalen, 1959); G. H. Pertz, ed., G. W. Leibniz. Gesammelte Werke, Part I: Geschichte, 4 vols. (Hannover,