VI.15 Gottfried Wilhelm Leibniz

b. Leipzig, Germany, 1646; d. Hanover, Germany, 1716

Calculus; theory of linear equations and elimination theory; logic

Renowned among mathematicians for his invention of the calculus, Leibniz was a universal thinker who graduated in law and was self-taught in mathematics. In 1676 he became counselor and librarian in Hanover for the Duke Johann Friedrich of Braunschweig–Lüneburg, holding this position until the end of his life. Besides mathematics he occupied himself with technical, historiographical, political, religious, and philosophical questions. His philosophy distinguished between two areas of reality: the world of appearances and the world of substances. It was in developing his philosophy that he was led to declare that the real world is “the best of all possible worlds.” In 1700 he was appointed first president of the newly founded Brandenburg Society of Sciences established in Berlin.

Gottfried Wilhelm Leibniz

Most of his mathematical ideas and writings were not published during his lifetime, and consequently many of his results were rediscovered many years later. About a fifth of his mathematical papers have now been published. He was always more interested in general or even universal methods than in technical details, using analogy and inductive reasoning to develop the art of invention. For the same reason he became a key creator of mathematical notation: he knew how much a suitable notation could facilitate mathematical discoveries.

One of Leibniz’s earliest mathematical works was a treatise on infinitesimal geometry (written in 1675–76 but not published until 1993). In it he used his “quanta” concept of the infinite. In Leibniz’s eyes the actual infinite as well as indivisibles, in the strictest sense of the word, were not quantities and therefore not mathematical entities: hence, he used the notions “infinitely small” and “infinitely large.” These denoted, it is true, variable quantities, but nevertheless they were quantities of a sort, so they could be handled by mathematics. Among the results in this treatise is a rigorous proof, in the style of ARCHIMEDES [VI.3], of the existence of (what is today known as) the RIEMANN INTEGRAL [I.3 §5.5] of continuous functions, which is based on intermediary values of the function within subintervals. Only a few of these results were actually published by Leibniz, and even these mainly without proof: in 1682 the alternating series for π / 4; in 1691 some further results. In 1713 he communicated his alternating series test in a private letter to JOHANN BERNOULLI [VI.18].

The year 1675 was also the year in which Leibniz invented his version of the differential and integral calculus, although its publication did not begin until 1684. His calculus was based on the key concept of a variable (quantity) ranging over a sequence of values infinitely close to each other, with the differential, the difference between two successive values in the sequence, being itself a variable that could be manipulated in the usual manner. Differentiation was represented by the operator “d”, which assigned variables to variables. For example, if x is a line of variable length, then dx is a very short line, also of variable length. Integration meant summation. His notation (d and ∫) is still used today. He deduced the standard differentiation rules (the chain rule, the product rule, etc.) and successfully applied his calculus to the differentiation of families of curves, to differentiation under the integral sign, and to various types of differential equations.

Leibniz considered “combinatorial art” as a general qualitative science, which did not coincide with modern combinatorial analysis but included combinatorics and algebra: Leibniz considered it as “the inventive part of logic.” Here he found the Girard formula for the representation of sums of powers of roots of equations by means of elementary symmetric functions, and the so-called Waring formulas by which polynomial symmetric functions are reduced to power sums (these were rediscovered by WARING [VI.21] in 1762). He invented double and multiple indices in order to solve systems of linear equations and problems of elimination theory. Between 1678 and 1713 he laid the foundations for the theory of DETERMINANTS [III.15]. The method now known as Cramer’s rule, for solving simultaneous equations, which in modern terms is based on determinants, and which Cramer published in 1750, was in fact found in 1684 by Leibniz (but again not published by him). He also stated (without proof) several theorems in the theory of linear equations and elimination theory now attributed to EULER [VI.19], LAPLACE [VI.23], and SYLVESTER [VI.42].

Among Leibniz’s other mathematical interests was additive number theory. In 1673 he found a recursion formula for the number of tripartitions of a natural number (published in 1976) and discovered further rules of recurrence now attributed to Euler. He also developed a formalism for a positional calculus (calculus situs) in order to express positions in space: if the definitions of figures are completely expressed by this calculus, all of their properties can then be found by this calculus. This is closely linked with the modern notions of geometry and topology.

Leibniz was one of the pioneers of actuarial theory. Using mathematical models of human life he calculated the purchase price of life annuities both for single persons and for groups of men, and he applied such considerations to the liquidation of a state’s indebtedness.

From the very beginning of his scientific career Leibniz was deeply interested in logic. He conceived of a general science: that is, of an art of inventing and of judging all sciences by means of sufficient data and a suitable universal language or writing. Yet, his “characteristica universalis” and the ensuing logical calculi remained fragmentary projects. His “calculus ratiocinator” was meant to be a formalized deduction of truth. Given that Leibniz was interested in formalizing calculations, it is not surprising that he also constructed the first four-function calculating machine. In constructing this machine he invented a new technical device, which he developed in two different versions: the so-called pinwheel (before 1676) and the stepped drum (from 1693 or earlier).

Further Reading

Leibniz, G. W. 1990-. Sämtliche Schriften und Briefe, Reihe 7 Mathematische Schriften, four volumes (so far). Berlin: Akademie.

Eberhard Knobloch