Leibniz's Theory of Elimination and Determinants

Abstract

As late as 1938, people doubted whether Gottfried Wilhelm Leibniz ever dealt with determinants. Thus, Gerhard Kowalewski wrote in 1938: "Strangely, nothing relating to determinants and their application has been found in his (viz. Leibniz's) manuscripts until now" [19, p. 125]. Later on, Morris Kline, hinting at Leibniz's often cited letter dating from 1693 to L'Hospital, erroneously wrote in 1972: "The solutions of simultaneous linear equations in two, three, and four unknowns by the method of determinants was created by Maclaurin, probably in 1729, and published in his posthumous Treatise of Algebra (1748)" [8, p. 606]. In 1972, the most important Leibnizian treatise on systems of linear equations appeared as Knobloch [9]. A long sequence of papers [10, 11, 12, 13, 14, 15, 16, 18] by Knobloch followed dealing with Leibniz's theory of elimination and determinants. Yet, all those papers remained partly unknown. In 2000 a historical survey of the evolution of algebra [1] appeared that again knew only Leibniz's letter to L'Hospital that was published in 1850 for the first time and added on 149f further false information about Leibniz's index notation. The booklet represented the state of the art of 1850. In the following paper I would like to summarize Leibniz's main ideas and results regarding determinants and elimination theory in order to demonstrate that Leibniz laid the foundation of the theory of determinants in Europe between 1678 and 1713, in other words, at the same time as his famous Japanese contemporary Seki. © Springer Japan 2013.