From Wallis to Newton: A way to calculus. Quadratures, series and infinite representations of transcendent quantities and forms (Q2702314)

This ambitious study, based on some essential, original texts issued in the field, aims at resuming ``un percorso di pensiero'' (i.e., a trajectory of thought), starting from the tentative of Wallis (in his attempt to subject to analysis certain transcendent quantities and forms) up to the establishment, by the great Newton, of some essential elements of calculus. NEWLINENEWLINENEWLINEThe possibility of translating a large class of geometrical problems and objects in terms of equations is discussed.NEWLINENEWLINENEWLINEThe analysis starts from Wallis's \textit{Arithmetica infinitorum} (published in 1655 in Oxford), the unique objective of which is that of establishing a quadrature method capable of demonstrating the quadrature of the circle.NEWLINENEWLINENEWLINEThe main sections of the study follow the structure of Wallis's book; thus, the first part is devoted to the analysis of the method of the indivisible objects, the second discusses the algorithm of quadrature for simple algebraical curves, while the third one deals with quadrature of the circle. There follows the presentation of Newton's involvement into such a fascinating problem. Inspired by Wallis, Newton analyzed the quadrature of the parabola and hyperbola (``to square the parabola'') and, later on, the discovery of the binomial development, based on resuming Wallis's main arguments. The results obtained by Newton, without constituting the fundamentals of calculus, provide essential elements for the elaboration of his new theory in the domain of tangents and curvatures.NEWLINENEWLINEFor the entire collection see [Zbl 0954.00008].