![]() Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. The Leibnizian law of continuity similarly finds echoes in Euler. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. ) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (. We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. Thus, one cannot infer the existence of infinitesimals from their successful use. ) by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions-a conception he later characterized as “syncategorematic”. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. These images from its George Arthur Plimpton Collection are presented through the courtesy of the Columbia University Libraries.įor additional images of Stevin's mathematics, see Mathematical Treasure: Simon Stevin's Oeuvres Mathematiques here in Convergence.įrank J.In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. ![]() Pages 440 and 441 provide exercises in performing operations involving decimal fractions. Here on pages 142 and 143, Stevin discussed his system for decimal fractions. This is the title page of L’Arithmetique. ![]() ![]() In this work, he reiterated his theory of decimal fractions, provided a unifying theory for quadratic equations, and introduced a method of approximation to find the roots of higher order algebraic equations. In 1585, he published L’Arithmetique, one of only two books that he wrote in the French language. Simon Stevin (1548–1620) was a Dutch military engineer, a mathematician, and an adventurer.
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