A GeoGebra Rendition of One of Omar Khayyam's Solutions for a Cubic Equation - Polynomials in History of Mathematics Courses

Deborah Kent (Drake University) and Milan Sherman (Drake University)

Since algebra is a foundational component of modern mathematics very familiar to undergraduates, it can provide a helpful touchstone throughout a history of mathematics course. Investigating the treatment of finding solutions for polynomial equations can provide a nice coherence to a range of material from ancient Greece through Niels Henrik Abel's nineteenth-century proof that no general solution exists for a quintic equation. Along the way, students encounter the work of Diophantus, al-Khwarizmi, Cardano, Viete, Descartes, Recorde, Newton, Euler, Lagrange, Gauss, Ruffini and many others as a notion emerges of algebra as a generalized problem-solving tool. This focus encourages students to think about how mathematical practitioners throughout history have dealt with similar kinds of problems. These investigations also can be used to highlight particular features of the historical context, perhaps a view of the purpose of mathematics, or the role of abstraction, or the nature of mathematical justification. The investigation of equation-solving also raises significant questions about the nature of notation and the development of algebra.  

While the modern framework of algebraic solutions to polynomial equations can aid student understanding, it can also hinder deeper appreciation for the nuance of historical mathematics in context. So familiar is the modern notion of a quadratic equation of the form \(ax^2 + bx + c = 0\) solvable by the quadratic formula, that it is easy to read modern ideas into an historical text. It is indeed a challenging mental exercise to think in a context that allows only positive integer coefficients – a place where \(x^2+ bx = c\) fundamentally differs from \(x^2 + c = bx\) – or to imagine a mathematical world where \(x^2\) literally represents a square and dimensionality fundamentally matters in a polynomial-like expression. The way historical practitioners approached these problems can feel very unnatural to modern students.

The demonstration here is designed to minimize these difficulties so students can first understand the result of Khayyam's construction before delving into historiographical questions. In particular, the GeoGebra applets connect Khayyam's geometrical approach to the cubic and a familiar cubic graph so that students can visualize the relationship between Khayyam's construction and the idea of \(x\)-intercepts as roots of the polynomial. The modern software furthermore allows a dynamic representation that enables students to explore this relationship for a variety of cubic equations. The modern technology provides scaffolding for students to understand Khayyam's construction and result. This knowledge will then support further historical discussion about what Khayyam's static image communicates about his own mathematical context.