A GeoGebra Rendition of One of Omar Khayyam's Solutions for a Cubic Equation - Conclusion

In his Treatise on Algebra, Khayyam set out to deal systematically with solving fourteen types of cubic equations using constructions and justification based on intersecting conic sections (Rashid and Vahabzadeh). The demonstration in this article illustrates his approach in one case of the cubic, the one we wrote as equation (0.16), \(x^3 + ax^2 + bx = c,\) with \(a, b, c > 0,\) in our list in the section, Omar Khayyam and Cubic Equations. Demonstrations of Khayyam’s solution methods for the cubic in equation (0.8), \(x^3 + ax = b,\) where \(a, b > 0,\) are available at GeoGebra via the following links:

• Maurice O'Reilly, Omar Khayyam's approach to solving certain cubic equations, https://www.geogebra.org/m/nbwq58f8
• John Golden, A Cubic Solution of Omar Khayyam, https://www.geogebra.org/m/hF8Hjuv6
• Paul Rakow, Omar Khayyam's geometric solution of a cubic equation, https://www.geogebra.org/material/show/id/rTV3y4Bb
• ank3721, Khayyam_NCTM, https://www.geogebra.org/m/YZvw7sux

Like the applets in this article, these all provide examples of how technology can facilitate student engagement with substantial questions about the history of mathematics.

The use of GeoGebra software reduces the difficulty of interpreting Khayyam's static diagram, and of dissecting the dependencies inherent in the construction to see how the technique can apply to every cubic of this form. Reflecting Khayyam's construction and superimposing a Cartesian graph of the cubic also provides a useful connection between his unfamiliar approach to solving a cubic and the more familiar process of locating a root on a graph. Tools like these GeoGebra applets frame historical results in a language accessible to students so they can first understand the result – but this is not the end of the story. The twenty-first century software provides scaffolding for deeper exploration of historiographical questions.

In a mathematical context where powers of \(x\) literally corresponded to geometrical dimensions, solving cubic equations represented the height of mathematical achievement. But what did Khayyam achieve? Given the above observation, is it accurate to say that Khayyam did in fact, solve the complete cubic problem? Or did Cardano do something unimagined by Khayyam? In what ways are or are not these techniques addressing similar questions? Questions like these invite students to delve into issues of mathematical and historical context. What does it mean to attain a mathematical result? Investigating solutions to cubic equations that do not persist as the first general solution gives rise to explorations of innovation and novelty. These GeoGebra representations may provide students with an understanding of Khayyam's marvelous construction that is adequate to engage these and other questions, facilitating fruitful discussion in an history of mathematics classroom.