Peano on definition of surface area

TL;DR

This paper explores the historical development of the concept of surface area through Peano's work, emphasizing the influence of Grassmann's calculus and Peano's measure derivatives, highlighting key approaches from the late 19th century.

Contribution

It analyzes Peano's evolving definitions of surface area and their connection to Grassmann's calculus, providing historical insights into the mathematical development of surface measurement.

Findings

Peano's geometric approach to surface area (1887)

Bi-vectorial approach to surface area (1888)

Influence on subsequent surface area theories

Abstract

In this paper we investigate the evolution of the concept of area in Peano's works, taking into account the main role played by Grassmann's geometric-vector calculus and Peano's theory on derivative of measures. Geometric (1887) and bi-vectorial (1888) Peano's approaches to surface area mark the development of this topic during the first half of the last century. In the sequel we will present some significative contributions on surface area that are inspired and/or closely related to Peano's definition.