Seminar 4: Diagrammatization of written mathematical practices

Written Calculation Practices and Numerical Notation in Fifteenth- and Sixteenth-Century Chinese Mathematical Texts

Célestin Xiaohan Zhou (Institute for the History of Natural Sciences, CAS, & School of Mathematics, The University of Edinburgh)

In general narratives of the history of mathematics in China, scholars usually focus on the fact that during the fifteenth and sixteenth centuries computational tools were in a transitional phase, shifting from counting rods to the abacus. However, a closer examination of mathematical writings from this period reveals that, for operations such as multiplication and division that had previously relied on counting rods, there existed a diversity of methods for computation as well as for determining the order of magnitude of the results. Mathematical texts such as Great Compendium of the Nine Chapters on Mathematical Methods with Analogies (Jiuzhang suanfa bilei daquan, 1450) and Unified Lineage of Mathematical Methods (Suanfa tongzong, 1592) introduced distinctive computational approaches by means of writing practices, including procedures such as “calculation with magic squares” (Hetu shushu) and “written calculation” (xiesuan). The prescriptions and descriptions of these procedures in these works further reflect the ways in which the authors diagrammatized their written mathematical practices. In these contexts, what is the status and meaning of notations in the diagrammatic configurations? How is the value of place manifested within such computations? What distinctions and relationships exist between the numerical notations used in the diagram and the verbal explanations that assign meaning to numbers within the context? In what ways do these practices differ from and connect with earlier rod-calculation practices and later abacus practices? During the process of rendering operations into visual diagrams, which operational or instrumental elements were symbolically recorded and incorporated into printed texts? These questions constitute the focus of the present presentation.

Typesetting Modern Algebra: The Modern Reception of Early Modern Contexts in Théotiste Lefèvre's Guide Pratique du Compositeur, 1855

J.P. Ascher (University of Edinburgh, SIGMA UKRI-ERC Postdoc)

Algebraic typesetting has an uneven history. While Joseph Moxon describes typography as a mathematical art in 1683-4 after he had likely produced some of his own type for abstract symbol systems, he provides little specific guidance to setting mathematics of any sort. In 1755, John Smith explains that most compositors dislike setting algebra because the authors invent their own symbols, are very particular, and don't understand the difficulties printing. Théotiste Lefèvre's manual of 1855, then, is one of the earliest systematic, published accounts of setting mathematics for a larger community of printers. But even more importantly, Théodore de Vinne sponsors a translation of Lefèvre by Henry Fine for his epoch-making manual of 1904 rather than writing a new section on algebraic typesetting. Lefèvre's approach, then, appears to become standard in the anglophone and francophone world, for example influencing the 1962 American Mathematical Society's Manual for Authors. This paper will present Lefèvre both retrospectively and prospectively: examining the particular early modern practices for typesetting that he aimed to regularise and showing how those practices became part of the look of modern algebra. In situating the practices chronologically, the paper attempts to think about how now-invisible, taken-for-granted aspects of modern mathematical symbolism can be investigated and articulated backwards. Influence is difficult to trace and this paper proposes to reflect on approaches to influence as well as laying out the historical facts.

Commentator: Aileen Fyfe (University of Saint Andrews)