Article ID: CBB986988543

Leibniz on the Requisites of an Exact Arithmetical Quadrature (2018)

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In this paper we will try to explain how Leibniz justified the idea of an exact arithmetical quadrature. We will do this by comparing Leibniz's exposition with that of John Wallis. In short, we will show that the idea of exactitude in matters of quadratures relies on two fundamental requisites that, according to Leibniz, the infinite series have, namely, that of regularity and that of completeness. In the first part of this paper, we will go deeper into three main features of Leibniz's method, that is: it is an infinitesimal method, it looks for an arithmetical quadrature and it proposes a result that is not approximate, but exact. After that, we will deal with the requisite of the regularity of the series, pointing out that, unlike the inductive method proposed by Wallis, Leibniz propounded some sort of intellectual recognition of what is invariant in the series. Finally, we will consider the requisite of completeness of the series. We will see that, although both Wallis and Leibniz introduced the supposition of completeness, the German thinker went beyond the English mathematician, since he recognized that it is not necessary to look for a number for the quadrature of the circle, given that we have a series that is equal to the area of that curvilinear figure.

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Authors & Contributors
M. Rosa Massa-Esteve
Harry R. Lewis
Mellado-Romero, Antonio
Linero-Bas, Antonio
Herrero-Piñeyro, Pedro J.
Gómez-García, Francisco
Concepts
Mathematics
Arithmetic
Geometry
Algebra
Physics
Calculus
Time Periods
17th century
18th century
16th century
19th century
Early modern
Modern
Places
Europe
China
Japan
Germany
France
British Isles
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