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.
Using archival sources, this article reconstructs little-known elements of the biography of Maurice Kraïtchik (1882-1957) and follows his activities from early work in Belgium to his exile in the United States. Particular attention is given to his role as editor of the Sphinx Revue and organizer of the Congrès International de Récréation Mathématique, which created new spaces for mathematical exchange. The article argues that Kraïtchik’s career illustrates the ambiguity between “amateur” and “professional” mathematics and shows how recreational problems, journals, and communities formed a significant part of twentieth-century mathematical culture.
This paper examines the use of dotted lines in the diagrams of early modern printed editions of Euclid’s Elements and argues that their changing roles are closely tied to shifting conceptions of quantity. Based on a corpus of sixty extant editions published before 1650, the study traces the distribution, functions, and transformations of dotted lines across different editions and editorial contexts. In the arithmetical sections of the Elements, dotted lines represent discrete units and instantiate specific numerical examples. However, their use encounters practical and conceptual constraints in cases involving large values or hypothetical constructions, leading editors to adopt alternative visual strategies. In later editions, dotted lines are increasingly supplemented or replaced by Hindu–Arabic numerals, supporting more abstract representations of numbers and the relations between them. In the geometrical books, dotted lines appear later and mark continuous quantities, serving a range of visual functions and becoming ubiquitous in the seventeenth century. Close analysis of Book X shows how different representational conventions are integrated within a single diagram. Methodologically, the paper combines close visual reading with a digital workflow for large-scale diagram extraction and classification, allowing a systematic analysis. The findings show that dotted lines function as a flexible visual resource whose changing uses reflect and support broader shifts in the conceptualization of quantity in early modern mathematics.
The aim of this paper is to give an overview of the theses defended by some of the mathematics students of Pierre Varignon, professor of mathematics at the Colle`ge des Quatre-Nations, Paris, from 1688 to 1722. They are to be discussed in the context of the genre of printed thesis booklets, with emphasis on the materiality and paratextual environment - title page, printer, dedications, odes, illustrations -, as well as on the structure and contents of the text. Finally, to give a concrete example, theses de architectura militari (1690) will be examined more in detail.
The study of the libraries of a social group provides valuable information on the evolution of the social group itself. In other words, the heritage of a School reflects its history by reproducing its chronology, activity and social dynamics. This paper offers the first historical analysis of the libraries of a scholarly mathematical community in Italy - the Italian School of Algebraic Geometry -, basing on a thoroughly exam of the personal book collections of some of its members (C. Segre, G. Fano, F. Enriques, F. Severi and A. Terracini), working between the 80-s and the second world war.
This article is devoted to the first general printed catalogue of the Ecole des Ponts et Chaussees (EPC) library, which was published in 1872 and whose production spanned several decades. The article opens with a historical account of the EPC library, and proceeds to examine the principal episodes that shaped the conception, maturation, and implementation of the 1872 catalogue, while its last part is concerned with the place of mathematical sciences-rational and applied mechanics included-within that document. Finally, the article's conclusion offers some general reflections on the process of designing and crafting library catalogues for engineering schools dedicated to the production of State engineers in 19th century France, while a special attention is given to the "patrimonialization" work carried out by those involved in the devising of the 1872 catalogue.
The mathematician Daniel Augusto da Silva is distinguished for his pioneering contributions to the scientific study of Portuguese pension funds providing survivors' pensions in the third quarter of the 19th century. The provision of this social protection in Portugal has been documented since the late 18th century, in montepios de sobreviv & ecirc;ncia. However, it was not until the 20th century that the foundations of such pension funds were based on the principles of actuarial science. This paper provides an analysis of da Silva's texts on actuarial calculus, placing them in the broader context of the development of this field in Portugal.
Between the years 2022 and 2024 Eberhard Knobloch and Karin Reich prepared and published the critical edition of the letter from Leibniz to James Bruce from November 21, 1712 and Leibniz's accompanying ,,Instruction". In this instruction Leibniz described a globe with declination lines, which was missing. In the beginning it was not known, whether the magnetical globe mentioned in these documents still existed. Detailed researches in Hannover and St. Petersburg until then had not been successful. On October 28, 2022 the globe Nr. 7, located in Gottingen was interpreted as the globe which Leibniz had described. This interpretation is due to Albert Krayer, who knew about the globe since 1986, but he did not know the text from Leibniz.
Descartes' Mesolabium method for solving cubic equations faced critiques by William Shea and John Schuster, who noted alleged errors in equation transitions and handling negative terms. Thus Descartes' approach was conceptually valid. Using substitutions like Vie`te's and Cardano's techniques, negative terms can be transformed, enabling Mesolabium solutions. While Descartes made superficial errors in private notes, deeper analysis shows his reduction method worked. Historical context explains avoiding negatives via mathods specific for the 16th and early 17th century, illustrating how past mathematical practices, though differing, maintained conceptual coherence.
The understanding of Leibniz’s conception(s) of the mathematical infinite has significantly improved in the last thirty years and Eberhard Knobloch’s work, as an editor and as a commentator, has played a crucial role in this evolution. My aim in this note is to assess how these studies have modified and improved our understanding of Leibniz’s views on infinity. I also indicate some avenues it opened for exploration and which are now the object of ongoing researches.
Jose Monteiro da Rocha (1734-1819), a seminal figure in Portuguese mathematics and astronomy, amassed a personal library that reveals the intricate cultural dynamics and circulation of scientific knowledge of the era. This article explores Monteiro da Rocha's collection, mapping how these books were crucial to his pedagogical and scientific work while examining the broader context of knowledge dissemination during the Enlightenment. As a key figure in the curricular Reform of the University of Coimbra in 1772, Monteiro da Rocha played a vital role in establishing the new Mathematics and Natural Philosophy faculties. His contributions to drafting the new statutes and his influence as an educator underscore the significance of his library, comprising over 1200 titles. The diversity of works in his library, including those by notable figures such as Newton, Euler, and Bernoulli, highlights the range of scientific knowledge circulating during the Enlightenment. Analysing this collection allows us to assess Monteiro da Rocha's impact on his teaching and scientific practice. Through an analysis of the content and provenance of these books, this article demonstrates how Monteiro da Rocha's library mirrors the scientific practices and priorities of the 18th century, while also serving as a testament to the cultural and scientific exchanges that defined the Enlightenment. Ultimately, research on Jose Monteiro da Rocha's library illuminates the practices of acquiring, reading, and using scientific texts, thereby significantly contributing to the intellectual and cultural history of this crucial period. By studying Monteiro da Rocha's library, we aim to shed light on the evolutionary landscape of scientific education in late 18th- and early 19th-century Portugal, emphasising the role of personal collections in shaping academic and cultural knowledge.
Giovanni Battista Benedetti (1530-1590) derived two constructions of ovals given their minor and major axes. These were published in 1585 and seem to be the first solution to this problem. Therefore, the generally accepted view that "the geometrical construction for drawing an oval for any given proportion was not known in the sixteenth century" (Ana L & oacute;pez Mozo, 2011; Nexus Network Journal, p. 571) has to be revised. We provide a transcription of Benedetti's Latin text and a translation, analyze his mathematical reasoning in detail, and review Benedetti's impact on the construction of ovals.
