Seminar 3: Diagrammatization of written mathematical practices

A diachronic look at diagrams

Jeffrey A. Oaks (Department of Mathematics and Computer Science, University of Indianapolis)

In this talk I will show a variety of examples of diagrams from arithmetic and algebra books in medieval Arabic, Latin, and Italian, and continuing through sixteenth-century Europe. In these books we find three kinds: some diagrams show one step in a process, such as in the multiplication of two numbers expressed in Indian notation; some express a relation, like an algebraic equation; and still others express a mathematical object, whether a fraction, a composite irrational root, an algebraic polynomial, or an amount expressed in terms of more than one denomination (like degrees, minutes, seconds or days, hours, minutes). There just might be enough in the examples I will show to begin thinking about continuity and differences in practice, especially as the techniques of calculation passed from Islamicate countries to Europe. 

Numerical Configurations with Diagrammatic Features in an Anonymous Commentary on Śrīdhara’s [Book] of Three Hundred [Verses] (Triśatī): Algorithms for Extracting Square and Cube Roots

Taro Tokutake (SPHERE, Université Paris Cité)

In this presentation, I aim to explore numerical configurations with diagrammatic features found in algorithms for extracting square and cube roots as presented in the Exposition on the [Book] of Three Hundred [Verses] (Triśatībhāṣya). Śrīdhara, a mathematician who flourished in India around 800 CE, composed a Sanskrit arithmetic treatise, the [Book] of Three Hundred [Verses] (Triśatī). Over thirty manuscripts of this treatise have been handed down to us, and some of them are accompanied by commentaries. The Exposition on the [Book] of Three Hundred [Verses] is one such commentary. This commentary survives only in a single complete manuscript (LD Institute, Ahmedabad, No. 1559), and no information about its date, place, or author is found there.

As many Sanskrit authors do, Śrīdhara presents the algorithms for extracting square and cube roots in a versified form and briefly outlines only their core steps. The commentator, on the other hand, interprets the algorithms given by Śrīdhara and presents the calculation procedures step by step, sometimes offering numerical configurations and sometimes not. These configurations indicate an intermediate arrangement of digits in the course of the algorithms, which are executed on a working surface that exists outside the running text. Building on these observations, the following questions arise: Can diagrammatic features be perceived in the numerical configurations? How do these configurations function in the execution of the algorithms? And how do the configurations relate the running text to the practice involving them? To address these questions, I offer a detailed study of the algorithms for extracting square and cube roots on the basis of the Exposition on the [Book] of Three Hundred [Verses].