leibniz

Differential and Integral Calculus

On the basis of geometrical considerations, Leibniz succeeded in the period between 1673 to 1676 in solving generally and universally the long-standing problems of the determination of areas, as well as of tangents, for arbitrary curvilinearly-limited (continuous) figures. To this end he made use of the similarity of the characteristic triangle and skillfully-selected finite triangles, of the complementarity of area and tangent determination and of the series expansions of rational terms. Thus he arrived at important integral transformations, as e.g.

at the Fundamental Theorem of the infinitesimal calculus

and at fundamental power series expansions, as e.g.

,

from which one obtains for x = 1 the so-called Leibniz Series. At the same time Leibniz introduced the notation that is still used today for the differential quotient and the integral sign.

The basic rules for differentiation he first published in 1684 in his celebrated paper Nova Methodus pro maximis et minimis. Indications of the integral calculus followed a few years later.

Further reading:   J.E. Hofmann, Leibniz in Paris 1672-1676 (Cambridge, 1974).

back to: Leibniz´ life and work

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