Delving into the History of Sciences, with a focus on Mathematics and its history, specifically on the History of Mathematics in Brazil, including its notable figures and works that significantly contributed to the development of this science, the present proposal aims to provide historiographical insights regarding the history of vector calculus in Brazil. Previous research data indicate that this subject, as a discipline, was first introduced in the country in 1926 at the Polytechnic School of São Paulo, with engineer-mathematician Theodoro Augusto Ramos (1895-1935) as the course instructor. Ramos was also the author of the first work published in the country (1927) and the first international publication by a Brazilian on the subject (France, 1930). These works have been analyzed and published, along with the academic and professional biography of Ramos, enriched with personal facts previously unknown to the history of Brazilian mathematics. The analysis of Ramos' two works revealed, among other aspects, a strong influence from the tradition of Italian authors, specifically Burali-Forti and Marcolongo. Moreover, throughout Ramos' publications, he mentions his correspondence and the topics discussed with prominent French figures of the time, such as Borel and Denjoy. In this regard, this work seeks to discuss the subsequent developments of vector calculus in Brazil within various contexts, focusing on the relationships established between the two countries during the first three decades of the 20th century. This is a qualitative research study, and the proposed methodological theoretical framework is based on Ricoeur's reflections on narrative hermeneutics (2007, 2010).

This work aims at discussing interrelationships between mathematics underlying temperaments and physics of musical instruments in the Renaissance. An overview at history of these instruments can lead to answers about the conceptual knowledge that different cultures had over what one call now acoustics. In the context of temperament, the early modern period saw the growing use of geometry as an instrument for solving structural problems in theoretical music, a change not independent from those that occurred in the conception of ratio in the context of theoretical music. This article intends to analyze how Renaissance theorists present the structure of musical instruments, revealing mathematical-scientific knowledge implemented in the instruments from this period. In other words, the idea is to analyze under a historical approach musical instruments as the materialization of conceptions of temperament. It is also intended to discuss the relationships between theory and practice in order to analyze how musical instruments were built as well as the role of geometry and arithmetic in the solution of structural problems in music and in transformation of conceptual structure underlying theoretical music, leading to the identification between ratio and number. In order to do this, it is necessary a bibliographical analysis of the following books:
“Sopplementi musicali” (Gioseffo Zarlino);
“L’harmonie Universelle” (Marin Mersenne);
“History of musical instruments” (Curt Sachs);
“Dialogo Della Musica Antica Et Della Moderna” (Vincenzo Galilei);
“Quantifying music: the Science of Music at the First Stage of Scientific Revolution 1580-1650” (H.F. Cohen);
“Acoustics: Historical and Philosophical Development” (R.B. Lindsay).

In this presentation, we provide a socio-economic historiographical approach, focusing on the development of Dynamical Systems and Chaos Theories. This approach is based on the Hessen thesis, which describes the relationship between the emergence and evolution of Newtonian Mechanics in the 17th century. The emergence of Chaos Theory in the 1970s, followed by its solid foundation in the 1980s, brought significant changes to the development of Dynamical Systems Theory. These changes influenced the conceptual framework of scientific discourse and sparked an intense philosophical debate, particularly regarding the notions of randomness and determinism. In recent decades, interest in the history of Dynamical Systems and Chaos Theories has grown. Various studies reveal historiographical approaches that focus mainly on key figures' biographies in the field, the evolution of its conceptual framework, and the controversies surrounding the development of relevant mathematical concepts. This focus consolidates an "internal" historiographical approach. Here, we propose a different interpretative and methodological framework that considers the social, cultural, economic, and technological factors that influenced and shaped the development of the theory. In this historical process, the advent and deployment of computers played a central role. Our interpretative schema adopts the Science–Technology–Industry triangle, prioritizing the role of socioeconomic factors in facilitating the theory's rapid growth. As part of ongoing research, our presentation provides historiographical evidence covering the period from Poincaré's work on the three-body problem to the late 1980s, when Chaos Theory became firmly established. This evidence supports the functionality of our interpretative schema and complements the existing historiographical approaches.