The paper provides an analysis of Giuseppe Vitali’s contributions to differential geometry over the period 1923–1932. In particular, Vitali’s ambitious project of elaborating a generalized differential calculus regarded as an extension of Ricci-Curbastro tensor calculus is discussed in some detail. Special attention is paid to describing the origin of Vitali’s calculus within the context of Ernesto Pascal’s theory of forms and to providing an analysis of the process leading to a fully general notion of covariant derivative. Finally, the reception of (...) Vitali’s theory is discussed in light of Enea Bortolotti and Enrico Bompiani’s subsequent works. (shrink)

The main interpretative challenge set by the Fundamenta Nova Theoriae Functionum Ellipticarum lies in Jacobi’s transformation theory upon which the entire theoretical edifice of the treatise depends. Unfortunately, Jacobi did not convey any indication of how he attained his general formulae for rational transformations of elliptic functions. He limited himself to providing a posteriori verification of the validity of his claims. The aim of this paper is precisely to describe the heuristic path by which in 1827 Jacobi succeeded in finding (...) these transformation formulae. The proposed historical reconstruction will hopefully shed new light upon the emergence in Jacobi’s work of the inversion process of elliptic integrals of the first kind and thus of the elliptic function sinamu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u$$\end{document} itself. (shrink)

The theory of exterior differential systems plays a crucial role in Cartan’s whole mathematical production. As he once recognized, all the germs of his subsequent work were contained there. Indeed, it provided him with powerful technical tools that turned out to be very useful in many different fields such as the theory of partial differential equations, the theory of infinite dimensional Lie groups (Lie pseudogroups) and differential geometry. Nevertheless, scarce attention has been paid to this area of historical research thus (...) far. Although authoritative scholars have investigated the foundation of exterior differential calculus in Cartan’s early papers, no specific analysis of Cartan’s subsequent works laying the foundations of what nowadays is known as the Cartan–Kähler theory has been yet provided. This article represents a first attempt to remedy this unsatisfactory state of affairs by focusing on Cartan’s work on Pfaffian systems at the very beginning of the past century. The analysis of Cartan’s relevant papers is preceded by a description of the historical context in which such contributions were conceived. In this respect, special emphasis will be put on some works by Engel and von Weber on Pfaffian systems, which laid the basis for the subsequent geometrical developments of Cartan’s theory of exterior differential systems. (shrink)

The article offers a general account of the genesis of the absolute differential calculus (ADC), paying special attention to its links with the history of differential geometry. In relatively recent times, several historians have described the development of the ADC as a direct outgrowth either of the theory of algebraic and differential invariants or as a product of analytical investigations, thus minimizing the role of Riemann’s geometry in the process leading to its discovery. Our principal aim consists in challenging this (...) historiographical tenet and analyzing the intimate connection between the development of Riemannian geometry and the birth of tensor calculus. (shrink)

In 1873, W. K. Clifford introduced a notion of parallelism in the three-dimensional elliptic space that, quite surprisingly, exhibits almost all properties of Euclidean parallelism in ordinary space. The purpose of this paper is to describe the genesis of this notion in Clifford’s works and to provide a historical analysis of its reception in the investigations of F. Klein, L. Bianchi, G. Fubini, and E. Bortolotti. Special emphasis is placed upon the important role that Clifford’s parallelism played in the development (...) of the theory of connections. (shrink)