Colin Maclaurin; Treaty of fluxions; Fluxional method; Derivative and integral history; history of differential and integral calculus.

In this paper we present our reflections on a two-volume study of Colin Maclaurin's A Treatise of Fluxions and its connections to the teaching of differential and integral calculus, published in 1742. It is intended to support the thesis that the conceptual approaches present in these books by Maclaurin can subsidize the organization of historical sequences associated with the calculus content, to be connected as a complementary approach to the textbooks, in the restructuring of the ways of processing the teaching of differential calculus and full-time in undergraduate courses in Mathematics. To develop the historical investigation, we examined the work, in the various aspects related to the construction of the history of differential and integral calculus, focusing our attention on the logical foundations of the fluxional method idealized by Isaac Newton and, later, improved and expanded by Colin Maclaurin. The epistemological analysis involved the concept of fluxion and its implications in the consolidation of the derived idea, properties, rules and demonstrations, which became known as the direct method of fluxion. From this approach on the study of derivatives through the fluxional method, aspects emerged concerning the formalization of the study on the notion of integral, its methods, rules and properties, known as the inverse method. At the end of the research, we concluded that the potentialization and conceptual identity of calculus based on the study of the fluxional method can serve as subsidies for the conceptual training of future Mathematics teachers, to achieve a comprehensive learning of this subject, which later can unfold in the improvement of the teaching and learning process of differential and integral calculus, in order to re-signify and complement the conceptual approach present in calculus books, and thus strengthen the understanding of the concepts necessary for learning analysis, in the study of differential calculus and integral part of undergraduate courses in Mathematics.