As métricas de cayley e seus desdobramentos (1857-1895)

Abstract

The objective of this thesis is to elucidate how the development of Cayley’s metrics took place, in addition to reconstructing the contexts of reception and circulation of the mathematical work, explicitly or implicitly, invoked in the dynamics of developments related to the notion of metric. Through this investigation, we aim to shed light on the development of the concepts necessary for the establishment of Cayley’s metrics in the 19th century. We examined the process by which these concepts were developed and how the mathematicians involved in research on metrics participated in this process. In addition to the reception of these by the mathematical community, especially British, French, German and Italian. For this, we ask ourselves how the genesis of Cayley’s concept of metric took place. Also, how it was received in the second half of the 19th century in these mathematical communities. To answer the first question, we analyze the birth conditions of the theory of invariants. The works of several British mathematicians were analyzed and the methodology of “text networks” helped us to understand the context of mathematical production. As the main work takes place in the context of this theory, we also study the construction of Cayley’s geometric vision about the algebraic work with the invariants of homogeneous polynomials. Where we find a close relationship with the work of mathematicians Plücker and Hesse, about the theory of conics in analytical form, with the use of homogeneous coordinates. From this context, we build a presentation of the sixth memory upon quantics, entering the text and clarifying the genesis of the construction of Cayley’s metrics. Reception was studied, in the English case, through the analysis of the main journals of that period, concluding that the main mathematician in relation to reception was Clifford. In the German case, we start from Klein’s article on non-Euclidean geometries, seeing the differences with Cayley’s approach. In France, we present how the interest in the subject took place in the periodicals and point out interesting aspects in Darboux’s work on analytic geometry. We also analyzed the first ten years of Battaglini’s journal, where we discovered references to Cayley’s metrics. Finally, we explain all the aspects in which this thesis contributes to the history of mathematics in terms of Cayley’s metrics.