ESHS/HSS Annual Meeting

Root-Extraction Diagrams Across Media: Al-Kāshī, Viète, and the Transition from Dustboard to Paper

• In Event: Diagrammatic Computations and Their Symbolic Aspects (III)

English Abstract

The early mathematical sources dealing with arithmetic using place-value notations often mention dust board computations. This terminology reflects the use of dust-covered writing surfaces, common in some mathematical traditions. Expressions such as “arithmetic of board and dust” (ḥisāb al-takht wa al-turāb) regularly appear in titles of early arithmetic treatises produced in both the Islamic world and South Asian traditions. Such a board allowed practitioners to erase or move numbers easily. The results of intermediate steps were thus regularly deleted. By contrast, on paper, these results remained fixed and visible, making the entire process explicit.
The increasing availability of paper eventually facilitated a shift toward paper-based computation, though the exact time of this shift remains uncertain. Comparative study of similar techniques across works from different periods offers clues about this shift. A particularly significant source is Jamshīd al-Kāshī’s (d. 1429) Miftāḥ al-ḥisāb composed in the early fifteenth century. Synthesizing the arithmetic, algebra, and mensuration knowledge of its time, it is renowned for its elaborate root-extraction tables and other notable contributions. Widely studied in Persian and Ottoman worlds, it became a canonical mathematical text. My presentation examines al-Kāshī’s computational tables to determine whether they represent a form of written computation or illustrate procedures carried out on another surface.
A method comparable to al-Kāshī’s root-extraction tables also appears in François Viète’s (d. 1603) De numerosa potestatum ad exegesim resolutione, edited and published by Marino Ghetaldi in 1600. By comparing al-Kāshī’s with related examples in Arabic and Persian mathematical texts and with Viète’s approach, this study seeks to clarify the nature of these diagrammatic representations and to show how different writing materials contributed to shape the diagrammatic form and execution of mathematical procedures.

Author

• Zehra Bilgin, University of Edinburgh & Istanbul Medeniyet University