This article examines the contexts, conventions, and functions of diagrams in ancient Greek practical mathematics. In particular it argues that they play a central cognitive role in a reader’s engagement with problem texts, despite the variation between papyri and individual problem texts. The diagrams serve as structured aids to scribal calculation, aid memory in complex problems, and allow for the verification of results.The article also presents a complete survey and detailed commentary on all known geometrical diagrams preserved in published Greek mathematical papyri, providing a new reference point for the study of diagrams in ancient Greek practical geometry.
Galois is one of the best-known figures in the history of mathematics, celebrated for his groundbreaking contributions to what is now called Galois theory and group theory, and also remembered for his tragic early death. However, many writings often oversimplify his work, focusing his scientific legacy on these aspects.In this paper, we explore several lesser-known extraordinary works of Galois. First, a different mathematical reading of Galois theory for algebraic equations, and an analysis of the wide and deep impact of the principle of pairing groups and invariants in the study of algebraic equations. Second, we provide a new and detailed explanation of the deep concepts and insights underlying Galois’s planned third memoir on abelian integrals. This includes his very likely implicit use of local uniformizing variables and his understanding of global periods for algebraic functions—ideas that anticipate key features of the later theory of Riemann surfaces. Third, a clarification of Galois’s little-known theory of ambiguity in transcendental analysis and its previously unrecognized close connections with later developments in the theory of Lie groups.Finally, we examine his bold and original views on mathematics and education, relate them to his extraordinary achievements as a universalist in a certain sense, and comment on valuable lessons for us today.
The present paper explores the relationship between generality and iteration in algorithms for squaring and cubing as presented in Śrīdhara’s Sanskrit arithmetic treatise, the [Book] of Three Hundred [Verses] (Triśatī), along with its anonymous commentary. I first examine the structure of each algorithm as stated in the base text. I then interpret and reconstruct the procedures based on the descriptions in the commentary. Finally, I explore a substantial difference in the formulation of the two algorithms and propose a hypothesis concerning this difference and Śrīdhara’s practice of generality. The present study suggests that the generality of the algorithms is underscored by the concise expression of their iterative nature. This iterative aspect is articulated through specific keywords and phrases that play a central role within the algorithms.
