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In this paper we investigate the evolution of the concept of surface area in Peano’s mathematical research, taking into account the main role played by Grassmann’s geometric-vector calculus and Peano’s theory on derivative of measures. Geometric (in Applicazioni geometriche, 1887) and bi-vector (in Calcolo geometrico, 1888) Peano’s approaches to surface area, culminating into the celebrated Peano’s paper Sulla definizione dell’area d’una superficie, presented by Casorati for publication on Rendiconti dell’Accademia dei Lincei in 1890 and re-proposed in Peano’s textbook Lezioni di analisi infinitesimale (1893), mark the development of this topic during the first half of the last century. Moreover we will present some remarkable contributions on surface area that are inspired and/or closely related to Peano’s definition.
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Gabriele H. Greco, Sonia Mazzucchi, Enrico M. Pagani, Peano on definition of surface area. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 27 (2016), no. 3, pp. 251–286