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

In a general History of Mathematics survey course, the work of Girolamo Cardano is a common focal point for studying a general solution to the cubic. Cardano's well-known solution to the cubic appeared in Ars Magna in 1545 and is often credited as the first publication of a general solution to cubic equations.

Figure 2. The title page of Cardano's Ars Magna in Convergence's Mathematical Treasures (image courtesy of Columbia University Libraries).

Students usually encounter Cardano's solution either with 3D manipulatives of the type described by William Branson (Branson), or in terms of an algebraic formulation as done by William Dunham (Dunham, pp. 133-146). Not only does Cardano's method offer a very satisfying intersection of algebraic and geometric methods, but the related historical narrative is rich with drama and intrigue, briefly outlined below (Fauvel and Gray, pp. 253-254).

Scipione del Ferro, a fifteenth-century mathematics professor at the University of Bologna, knew a method of algebraic solution for equations of the form “cube and things equal to numbers,” or \(x^3 + ax = b\) with \(a, b > 0\). At some point, del Ferro shared his guarded knowledge with his student Antonio Maria Fióre, who proceeded to make a living by challenging others to mathematical problem-solving contests. Since Fióre knew del Ferro's secret, he always posed problems of the form \(x^3 + ax = b\).  Fióre's trick worked until he challenged Niccoló Tartaglia in 1535.

Figure 3. Niccoló Tartaglia (1499-1557), from Convergence's Portrait Gallery. (This image was provided by the Dibner Library of Science and Technology, The Smithsonian Institution Libraries, and its use must conform to the Library’s rules and standards.)

Tartaglia developed his own solution and won the contest (although he declined Fióre's prize of 30 free dinners). Word of Tartaglia's fame and accomplishment reached Girolamo Cardano in Milan. After a long campaign, Cardano convinced Tartaglia to meet with him in 1539 to divulge the secret in verse on the condition that Cardano swear not to publish it. Cardano and his student Ludovico Ferrari subsequently learned that del Ferro knew the solution method. Since they also developed the material further, discovering how to solve other types of cubics, and, thanks to Ferrari, also quartics, they decided to publish this work in Ars Magna in 1545. The enraged Tartaglia fought bitterly against Cardano and Ferrari for years afterwards.

Figure 4. Girolamo Cardano (1501-1576), from Convergence's Portrait Gallery.

The hint Cardano initially got from Tartaglia worked only for a special class of cubics of the form \(x^3 + ax = b\) with \(a, b > 0\). For contemporary readers, of course, \(ax^3 + bx^2 + cx + d = 0\) encompasses all possible cases, where the coefficients are positive, or negative, real, or imaginary. Prior to the widespread adoption of negative numbers, though, mathematical practitioners considered each possible combination of positive coefficients for a cubic equation. It is, naturally, difficult (if not impossible) for students to imagine practicing mathematics without the generalized problem-solving tools of modern algebra.  It is likewise often challenging for students to appreciate how, in context, Tartaglia had actually given Cardano the solution to one type of cubic. Cardano then continued working with Ferrari to extend the result to include other types of cubics. In total, they investigated fourteen types of cubic equations. Due to these efforts, Cardano's name is now attached to the three equations that give the three roots of the modern cubic equation \(ax^3 + bx^2 + cx + d = 0\), which encompasses all the various forms of medieval cubics.

Although Cardano gets significant credit, he and his fellow Italian mathematicians were certainly not the first to grapple with solutions of cubic equations. Some ancient Babylonian tablets from the 20th-16th centuries BCE have tables for calculating cubes and cube roots, although there is no evidence they used these tables to solve equations. The doubling of a cube is a well-known problem of classical antiquity attempted by mathematical practitioners including Hippocrates, Menaechmus, and Archimedes, although it is extremely unlikely that any of them formulated the problem in terms of a cubic equation (Aminrazavi and Van Brummelen). The Nine Chapters on the Mathematical Art, a classic Chinese text compiled around the 2nd century BCE, includes some methods for solving some cubic equations. In 7th century China, Wang Xiaotong solved 25 cubic equations of the form \(x^3+px^2+qx=N\) (Mikami, pp. 53-56). In the 11th century, Omar Khayyam contributed a geometric solution to the theory of cubic equations. In twelfth-century India, Bhaskara II attempted the solution of cubic equations. In Persia, Sharaf al-Din al-Tusi treated several types of cubic equations in his Treatise on Equations. The treatment of cubics additionally occupied many other mathematical practitioners from the 9th through the 16th centuries (Aminrazavi and Van Brummelen).

These considerable efforts are often overlooked in undergraduate history of mathematics survey classes. There are many defensible explanations for this, ranging from lack of space in a semester syllabus to the challenges – linguistic, cultural, and mathematical – posed by the material itself. The GeoGebra applet included here aims to highlight one of these seldom-seen investigations in an accessible way. Omar Khayyam's constructive solution to the cubic serves as one example from the richly textured history of work on finding solutions to cubic equations. The use of GeoGebra software removes the difficulty of interpreting Khayyam's static diagram, enabling students to focus on appreciating the construction and understanding the justification that the constructed segment is, in fact, a solution. This dynamic visualization also links Khayyam's construction with a modern Cartesian graph of a cubic equation. Students can interact with this dynamic representation, making Khayyam's remarkable work more accessible and engaging, and thus easier to integrate into a History of Mathematics course.