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

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

Khayyam's full name, Abū’l-Fath Ghiyāth al-Dīn ‘Umar ibn Ibrāhīm al-Khayyāmī al-Nīshāpūrī, suggests that his family trade was making tents, but his modern reputation hinges on the poetry, mathematics, and philosophy he generated in a region near present-day Afghanistan. Biographical information on Khayyam is taken from the article about him in the Dictionary of Scientific Biography (Rosenfeld and Youschkevitch).

Khayyam may be most well-known for his Rubā’iyāt quatrains which critique the fundamental tenets of religion through discussion of issues such as epistemology, eschatology, determinism, and quest for meaning. In mathematics, Khayyam worked within an Islamic tradition of investigating Euclid's parallel postulate and discussing the definition of ratios. In addition to producing a thorough investigation of cubics, Khayyam also discovered a general method for root extraction of arbitrarily high degree. He appears to have arrived at questions about cubic equations through his investigation of geometrical problems.

In a translated excerpt of Khayyam's work, he discussed a “square square,” or, for modern readers, an \(x^4\) term, as something that “does not exist in reality in any way,” but instead in the realm of philosophy (Fauvel and Gray, p. 226). He continued to say “whatever is obtained by algebra is obtained” by four things:  “number, object (\(x\)), square (\(x^2\)), and cube (\(x^3\)).”  Dimensionality provided the justification for this. Khayyam specified that number is “a state of mind independent of all magnitudes. This does not exist in reality.” Khayyam clarified that “number only comes into existence when it is denoted by a material cause.” For Khayyam, a straight line denoted the object familiar to modern readers as \(x\). A square, which we denote by \(x^2\), Khayyam “denoted by a quadrilateral of equal sides with right angles whose side is equal to a straight line.” And, finally, he described a cube, known to us as \(x^3\), as “a solid which is bounded by six equal surfaces of four sides whose sides are equal, angles are right angles.” Those four sides were each straight line objects, or, for us, \(x,\) and each surface was a square; in our notation, \(x^2\). Khayyam referenced Euclid's Elements (XI, 27) for the construction of a cube and asserted that “an object with more than three dimensions is impossible” (Fauvel and Gray, p. 226).

Khayyam stated that methods for finding unknowns for the six different forms of quadratic equations have been thoroughly “explained in books of algebraists” (Fauvel and Gray, p. 226). In fact, ruler and compass constructions to solve quadratic equations and similar methods date as far back as the Greeks. Khayyam would have been familiar with both the Elements of Euclid and the Conics of Apollonius. Khayyam enumerated the following types of cubic polynomials listed as equations (0.1) to (0.19) below. He used the language of “a cube equal to squares” for what the modern reader will recognize as \(x^3 = ax^2\), or “a cube plus edges equal to squares and numbers” for \(x^3 + ax = bx^2 + c,\) and so on, but we adopt polynomial notation here.

(0.1)   \(x^3\) \(= ax^2\)  
(0.2)   \(x^3\) \(= ax\)  
(0.3)   \(x^3\) \(= a\)  
(0.4)   \(x^3 + ax^2\) \(= b\)  
(0.5)   \(x^3 + ax^2\) \(= bx\)  
(0.6)   \(x^3 + a\) \(= bx\)  
(0.7)   \(x^3 + a\) \(= bx^2\)  
(0.8)   \(x^3 + ax\) \(= b\)  
(0.9)   \(x^3 + ax\) \(= bx^2\)  
(0.10)   \(ax^2 + bx\) \(= x^3\)  
(0.11)   \(ax^2 + b\) \(= x^3\)  
(0.12)   \(ax + b\) \(= x^3\)  
(0.13)   \(x^3\) \(= ax^2 + bx + c\)  
(0.14)   \(x^3 + ax\) \(= b + cx^2\)  
(0.15)   \(x^3 + ax^2 + b\) \(= cx\)  
(0.16)   \(x^3 + ax^2 + bx\) \(= c\)  
(0.17)   \(x^3 + ax^2\) \(= bx +c\)  
(0.18)   \(x^3 + ax\) \(= bx^2+c\)  
(0.19)   \(x^3 + a\) \(= bx^2 + cx\)  

Because the coefficients \(a\), \(b\), and \(c\) must be greater than zero for Khayyam, these represented different forms of cubic equations. Equations (0.1), (0.2), (0.5), (0.9), and (0.10) are solvable by methods from Euclid's Elements, Book II. Finding solutions for the remaining fourteen cubic equations required solid geometry, specifically conics and conic sections.

To contemporary students, a geometric solution to a cubic equation may seem strange. Exploring this approach poses a challenge of communicating how geometric problems motivated the study of cubic equations. Medieval Islamic algebra is, in many ways, a foreign terrain both conceptually and notationally. We do not have access to sources that reveal the particular insight that led to Khayyam's construction. In fact, this is a common situation when dealing with historical mathematics. Rarely do we have access to the developmental process of historical practitioners of mathematics. One exceptional example is the recent discovery of a previously unknown text, The Method, by Archimedes, that resolved centuries of questions about how he generated the formulas he proved by methods of contradiction. For now, we have no such insight in the case of Khayyam. This foray into medieval Islamic mathematics nonetheless invites discussion of the mathematical environment and tools available to Khayyam – as well as how they differed from those of Cardano and others. Encountering the perspective that powers of \(x\) correspond to actual geometrical dimensions is also a valuable addition to a student's experience of historical mathematics. In this context, bounded in ways by three dimensions of physical space, the solution of a cubic equation marked a significant achievement.