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

    attempts that were not recognized as fruitless because of the computational difficulties involved. On the other hand, Leibniz succeeded in demonstrating the universal validity of Cardano's formulas for solving cubic equations even when three real solutions are present and in establishing that in this case the imaginary cannot be dispensed with. The generality of these results had been frequently doubted because of the influence of Descartes. Through this work Leibniz concluded that the sum of conjugate complex expressions is always real (cf. the well-known example ?1 + ?-3 + ?1 - ?-3 = ?6). The theorem named for de Moivre later proved this conclusion to be correct.

In the fall of 1675 Leibniz was visited by Tschirnhaus, who, while studying Descartes's methods (which he greatly overrated), had acquired considerable skill in algebraic computation. His virtuosity aroused admiration in London, yet it did not transcend the formal and led to a mistaken judgment of the new results achieved by Newton and Gregory. Tschirnhaus and Leibniz became friends and together went through the unpublished scientific papers of Descartes, Pascal, and other French mathematicians. The joint studies that emerged from this undertaking dealt with the array of differences and with the “harmonic” sequence ..., 1/5, 1/4, 1/3, 1/2, 1/1 and was treated by Leibniz as the counterpart of the arithmetic triangle. They then considered the succession of the prime numbers and presented a beautiful geometric interpretation of the sieve of Eratosthenes—which, however, cannot be recognized from the remark printed in 1678 that the prime numbers greater than three must be chosen from the numbers 6n ? 1.

When Roberval died in 1675 Leibniz hoped to succeed him in the professorship of mathematics established by Pierre de la Ramée at the Collège de France and also to become a member of the Académie des Sciences. Earlier in 1675 he had demonstrated at the Academy the improved model of his calculating machine and had referred to an unusual kind of chronometer. He was rejected in both cases because his negligence had cost him the favor of his patrons. Nevertheless, his thorough, critical study of earlier mathematical writings resulted in important advances, especially in the field of infinitesimals. He recognized the transcendence of the circular and logarithmic functions, the basic properties of the logarithmic and other transcendental curves, and the correspondence between the quadrature of the circle and the quadrature of the hyperbola. In addition, he considered questions of probability.

In the late autumn of 1675, seeking a better understanding of Cavalieri's quadrature methods (1635), Leibniz made his greatest discovery: the symbolic characterization of limiting processes by means of the calculus. To be sure, “not a single previously unsolved problem was solved” by this discovery (Newton's disparaging judgment in the priority dispute); yet it set out the procedure to be followed in a suggestive, efficient, abstract form and permitted the characterization and classification of the applicable computational steps. In connection with the arrangement in undetermined coefficients, Leibniz sought to clarify the conditions under which an algebraic function can be integrated algebraically. In addition he solved important differential equations: for example, the tractrix problem, proposed to him by Perrault, and Debeaune's problem (1638), which he knew from Descartes's Lettres (III, 1667); and which required the curve through the origin determined by

dy/dx = x - y/a?

He established that not every differential equation can be solved exclusively through the use of quadratures and was immediately cognizant of the far—reaching importance of symbolism and technical terminology.

Leibniz only hinted at his new discovery in vague remarks, as in letters to Oldenburg in which he requested details of the methods employed by Newton and Gregory. He received some results in reply, especially concerning power series expansions—which were obviously distorted through gross errors in copying—but nothing of fundamental significance (Newton's letters to Leibniz of June and October 1676, with further information on Gregory and Pell supplied by Oldenburg). Leibniz explained the new discovery to him personally, but Tschirnhaus was more precisely informed. He did not listen attentively, was troubled by the unfamiliar terminology and symbols, and thus never achieved a deeper understanding of Leibnizian analysis. Tschirnhaus also had the advantage of knowing the answer, written in great haste, to Newton's first letter, where Leibniz referred to the solution of Debeaune's problem (as an example of a differential equation that can be integrated in a closed form) and hinted at the principle of vis viva. Leibniz also included the essential elements of the arithmetical quadrature of the circle; yet it was derived not by means of the general transmutation but, rather, through a more special one of narrower virtue. Tschirnhaus did not know that the preliminary draft of this letter contained an example of the method of series expansion through gradual integration (later named for Cauchy [1844] and Picard [1890]) and, in any case, he would not have been able to understand and fully appreciate it. On the other hand, he did see and approve the definitive