ESHS/HSS Annual Meeting

The Problem of Apollonius in the Notes of Thomas Harriott (c.1560-1621)

• In Event: Mathematics in Manuscripts 1

English Abstract

In his lost work Tangencies, the ancient Greek geometer Apollonius of Perga showed how to construct a circle tangent to three other objects, which could be points, straight lines, or circles. This problem captured the interest of many mathematicians, including the Early Modern French mathematician François Viète, who provided a ruler-and-compass solution. His English contemporary Thomas Harriot took many notes on the problem and on Viète’s solution, which are the focus of this study. In the notes, Harriot realized that the number of solutions of the problem depended on the relative positions of the three initial objects. Harriot’s rough attempts at classifying the number of solutions given the initial configuration of the problem display an intuitive understanding of the connection between topological properties and the field of enumerative geometry. Harriot’s geometrical sketches, in which he tried to list all possible configurations and to indicate the location of the solutions schematically, display a striking resemblance to those of mathematicians tackling the same issue at the end of the 19th century. The analysis of these notes shows the importance of the study of personal mathematical manuscripts, as these allow us to understand the creative process at the core of mathematical practice. Harriot’s folios force us to reconsider the linearity of the evolution of mathematical ideas, making clear how intuitions and experiments, which are not to be found in published works, lie at the core of the pursuit of mathematics, sometimes anticipating in embryonic form future developments.

Author

• Stefano Farinello, University of Hamburg