


































Global Research in Higher Education 
ISSN 2576-196X (Print) ISSN 2576-1951 (Online) 

Vol. 6, No. 2, 2023 

www.scholink.org/ojs/index.php/grhe 

32 

 

Original Paper 

An Important Historical Milestone: The Classification of the 

Cubic Equations 

Ilhan M. Izmirli
1
 

1
 George Mason University, Fairfax, VA, USA 

 

Received: January 11, 2023       Accepted: January 23, 2023    Online Published: April 13, 2023 

doi:10.22158/grhe.v6n2p32                URL:http://dx.doi.org/10.22158/grhe.v6n2p32 

 

Abstract 

This article investigates the use of the history of mathematics as a pedagogical tool for the teaching 

and learning of mathematics, using the history of the cubic equation as a specific example. 

Cubic equations arise intrinsically in many applications in natural sciences and mathematics. For 

example, in physics, the solutions of the equations of state in thermodynamics, or the computation of 

the speed of seismic Rayleigh waves require the solutions of cubic equations. In mathematics, they are 

instrumental in solving the quartic equations, for in the process, these are reduced to cubic equations. 

The impossibility of trisecting an angle or doubling a cube using only a straightedge and compass is 

equivalent to solving some cubic equations. As the name implies, the cubic spline approximation, an 

important tool in numerical analysis, also entails working with cubic functions.  

Although cubic equations were explored by the ancient Babylonian, Greek, Chinese, Indian, and 

Egyptian scholars, it took the collective work of many well-known mathematicians such as Diophantus, 

Archimedes, Fibonacci, del Ferro, Khayyam, Tartaglia, Cardano, Viète, Descartes, and Lagrange to 

finally obtain a full solution.  

Our goal in this paper is to investigate one of the most formidable steps in this extensive and prolific 

history, namely the complete classification of the cubic equations by Omar Khayyam in eleventh 

century, who was the first scholar to classify cubic equation and hence facilitate a methodical and 

logical approach to obtaining a general solution. 

Keywords 

Cubic equation, monic polynomial, classification of cubic equations 

 

 

 

 



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1. Introduction 

History of mathematics is more than a few amusing and enjoyable anecdotes and/or some unverifiable 

apocrypha. Indeed, it is an account of the process of development of the discipline over long periods of 

time and in different places using a variety of valid ways of constructing concepts and proofs. As such, 

it fosters flexibility and creativity in the learners, and provides them with crucial sources of inspiration, 

insight, and motivation. It also presents mathematics within a context that corroborates the notion that 

this discipline is a dynamic, continuously transmuting field of study that is open to refutation and 

revision; in other words, it humanizes mathematics (Bidwell, 1993). 

It is now well accepted that inclusion of the history of mathematics in the classroom not only refines 

problem-solving skills and helps students make useful mathematical connections (Jankvist, 2009), but 

it also emphasizes the interaction between mathematics and society (Wilson & Chauvot, 2000).    

To sum up, history, introduced in a dynamic and vigorous manner, has an essentially fundamental role 

to play in today’s mathematics classrooms. It demystifies and clarifies mathematics by showing that it 

is the creation of human beings and thus broadens the knowledge that students construct in a 

mathematics class (Marshall & Rich, 2000).  

 

2. Methodology 

To accentuate these points, in this paper, we will deal with a specific example: the historical 

development of the solution of the cubic equation. 

There are different ways of using history in the classroom (Bidwell, 1993). For example, we can inject 

anecdotal material as the course is presented, that is, we can make historical references to coursework 

while it is being covered. Our methodological approach is somewhat different: we aim to make the 

accurate historical developments of a topic (namely the categorization of the cubic equation) a part of 

the course (algebra or pre-calculus). This is what the example given in this article is trying to achieve.  

As is well known, today, using basic algebraic tools, the roots of the cubic equation 

 

can easily be computed. In fact, using Cardano’s formula we have 

 

 

 

In case an approximate solution is needed, we let 

 



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and apply Newton-Raphson recursion formula 

 

that is, we iterate 

 

until , where  is a given tolerance. 

In this paper, we are interested in one particular stage in the long journey that started with some 

cuneiform tablets and ended up with not only the general solution techniques, but also the invention of 

complex numbers: specifically, the important step of classification of the cubic equations. 

 

3. A Brief History 

Cubic equations were known to the ancient Babylonian, Greek, Chinese, Indian, and Egyptian 

mathematicians (Van der Waerden, 1983). Indeed, Babylonians had extensive tables for calculating 

cubes and cube roots (Cooke, 2012).  

The Greek involvement with the cubic equations grew out of their desire to solve the problems of 

doubling of the cube and trisection of an angle (Kline, 1990). 

Archimedes (c. 287 BCE-c. 212 BCE) in his On the Sphere and Cylinder II, dealt with the doubling of 

the cube problem and reduced the problem to the solution of some cubic equations of the type 

 

and 

 

(Katz, 2004). 

It was also shown that the angle trisection problem could be written as 

 

or equivalently as 

 

another cubic equation. 

In the 3rd century AD, Diophantus (200-284), in his famous book Arithmetica, proposed the following 

interesting problem: 

Find a right-angled triangle such that its area added to one of the perpendicular sides makes a square, 

while its perimeter is a cube. 

It was shown by Diophantus that the solution of this problem required finding the solution of the 

equation 

 



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Diophantus found the solution to this equation to be  (Heath, 2009).   

We also know that in the 7th century, the Chinese mathematician Xiaotong solved some cubic 

equations of the form 

 

(Van der Waerden, 1985). 

However, it should be noted that all of the above solutions were based on ad hoc arguments and 

applied only to specific problems and specific equations. No attempts were made to obtain a general 

method for solving the cubic equation. 

This all changed in the 11th century, when the Persian poet-mathematician-astronomer-philosopher, 

Omar Khayyam (1048-1131), made significant progress in the theory of cubic equations. First of all, he 

realized that a cubic equation could have more than one solution. Secondly, in his Fil 

Birâhini-al-el-mesâil-el-Cebr vel Mukabele, he wrote a complete classification of cubic equations. 

Thirdly, in the same book, he proposed some general geometric solutions that involved intersecting 

certain conic sections (Katz, 2004).   

Khayyam’s general geometric solution of the cubic as well as his contributions to literature, philosophy, 

astronomy, geometry, and sciences will be analyzed in detail in another paper. Here, we are only 

interested in his classification of the cubic equation. For, it is this classification that provided the much 

needed background for the works of Fibonacci (1170-1250), [19], Scipione del Ferro (1465-1526), 

Niccolò Fontana Tartaglia (1500-1557), Gerolamo Cardano (1501-1576), François Viète (1540-1603) 

and René Descartes (1596-1650) (Note 1) en route to the complete solution of the cubic.  

 

4. Omar Khayyam’s Life and Work  

Omar Khayyam (whose full name was Ebu Hafs Omar Ibn-i Ibrahim el Khayyami) was born in 

Nishapur in northeastern Iran in 1048. 

Khayyam came from a middle-class family. The patronymic Khayyam (or Khayyami) meaning “tent 

maker” implies that at least one of his ancestors must have been a tent maker. 

He was educated in Nishapur by some of the leading scholars and scientists of time such as 

Nasȋr-ed-din Sheih Muhammad Mansur and Muvaffak-üd-din Abdullatif Ibn-ül Lübâd, and the 

renowned theologian Hâce Ali. 

He died in the same city in 1131. For more information on his life and major works, please see Dilgan 

(1964). 

Khayyam was introduced to Western world in the first half of nineteenth century by the translations of 

two of his major works. One was an 1851 translation of his work in algebra (Fil 

Birâhini-al-el-mesâil-el-Cebr vel Mukabele) by Franz Woepcke (1826-1864) as L’Algèbre d’Omar Al 

Khayyamȋ and the other was the 1859 translation of his poetry (Rubaiyat) by the well-known English 



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poet and writer Edward FitzGerald (1809-1883) as The Rubaiyat of Omar Khayyam. His philosophical 

thoughts have been compared to Epicure, Voltaire, and Schopenhauer (Dilgan, 1964). 

Khayyam’s Fil Birâhini-al-el-mesâil-el-Cebr vel Mukabele comprised five sections. The first section 

contained a preface and gave some basic definitions. In the second section, Khayyam discussed first 

and second degree equations and in the third, he talked about cubic equations. Fourth and fifth sections 

involved equations that contained powers of unknown in the denominator and some additional remarks 

on basic rules of algebra.  

It must be mentioned that Khayyam, like his contemporaries, considered only equations whose 

coefficients and roots were positive numbers and if an equation did not have positive roots, he 

classified them as unsolvable.  

It is interesting to note that for the first time, Khayyam established some relationships between roots 

and the coefficients of an equation.  For example, in the equation 

 

where , he showed that there would be two positive roots if  and no positive roots 

(i.e., in his terms, no solutions) if .   

In the case  , he showed that the equation may or may not have positive roots. Indeed, 

some simple examples show this to be the case. For example, it can easily be verified that with 

, the equation has no positive roots; with , it has one positive root; 

and with , it has two positive roots.  

  

5. Khayyam’s Classification of the Cubic 

In section three of Fil Birâhini-al-el-mesâil-el-Cebr vel Mukabele, Khayyam divided cubic equations 

into four major groups.   

The first group consisted  equations. Only two of those could not be solved by the geometric 

method Khayyam proposed and required other techniques. Thus,  could be solved by Khayyam’s 

method.   

The second group contained trinomial equations that had three successive powers ( , and constant 

term or , and ). There were  such equations and these were all solvable by Khayyam’s 

method. 

The third group contained  equations that contained , and  terms, but not necessarily 

successively—the only stipulation was that a cubic term had to be present. All equations in this group 

were solvable by Khayyam’s method. 

The fourth and the last group contained  equations with four terms containing , and  in a 

successive manner. 

 



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Thus, Khayyam’s method applied to  

 

different type of cubic equations. 

Below we give the members of these groups. Here,  are positive constants. 

Group I 

 

 

 

 

 

 

 

 

 

 

(not solvable by Khayyam’s method) 

 

 

Group II 

 

 

 

 

 

 

 

 

 



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Group III 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Group IV 

 

 

 

 

 

 

 

 

 

 



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Let us give a specific example of Khayyam’s method. As mentioned earlier, the general method will be 

analyzed in a different paper. 

Say, for example, we want to solve the first equation of Group III 

 

Of course, since ,  cannot be zero.   

Khayyam’s solution consisted of constructing the parabola  

 

and the circle with center at  and radius  and determine the -coordinate of the intersction. 

For, if we do that, we get  

 

 

The second equation becomes 

 

that is, 



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Since  this gives us 

 

The root found by this method is the real and positive root since the length of a line segment cannot be 

negative or imaginary. 

For example, to solve 

 

we graph the parabola  and the circle  

 

 

 

and note that the -coordinate of intersection of the two curves for  is at , which of 

course is the only real positive solution of the equation. 

 

6. Conclusion 

Were one to choose to adhere to the instrumentalist or mechanical view of mathematical knowledge 

and instruction, one would then equate computational proficiency with mathematical understanding. In 

other words, in this particular case, one would just give Cardano’s formula (with or without proof) and 

use it to solve a few cubic equations. This, of course, would deny the students the true mathematical 

understanding of the problem: all of a sudden, somehow, a person discovered a formula, and all one has 

to do now is to substitute the right numbers in the right places, and then rely on the calculator. The 

instructor, with this approach, clearly would fail to provide the much needed insight to and motivation 

for the topic that is being covered.    

 



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In contrast, the historical approach, places problem solving at the heart of instruction. For instance, in 

our specific example, the students would realize that the solution of the cubic equation was developed 

over time and through the efforts of several scholars of different ethnic and cultural backgrounds. The 

reasoning behind the classification, the gist of our paper, would pave the path to students’ ability to 

indulge in the study of a class of objects rather than a specific object. By taking a historical approach to 

the subject, students would learn that the classification of the cubic was indeed a mathematical 

necessity on the way to the discovery of the general solution.   

As an added benefit, an analysis of the historical development of the solution of the cubic shows the 

internationalist character of mathematics. As teachers, if we are, as we should be, concerned about 

cultural chauvinism and parochial nationalism, and try to instill an unbiased perspective of mathematics, 

the historical approach is the needed panacea to convince learners that transnational collaboration, 

universal solidarity, and fellowship of scholars have always been the epitomes of mathematical 

development.   

It also shows the interconnectedness of mathematical areas (geometry, algebra, etc.). Moreover, it 

proves that even masters were prone to erroneous thinking (denial of negative solutions) and hence 

shows the dangers of taking mathematics as a set of absolute truths as opposed to a set of fallible and 

transmutable ones. 

 

References 

Bidwell, J. K. (1993). Humanize Your Classroom with The History of Mathematics. Mathematics 

Teacher, 86, 461-464. https://doi.org/10.5951/MT.86.6.0461 

Cooke, R. (2012). The History of Mathematics: A Brief Course (3rd ed.). John Wiley & Sons.  

Dilgan, H. (1964). Șair Matematikçi Ömer Hayyam (Poet Mathematician Omar Khayyam). Sirketi 

Mürettibiye Basimevi, Istanbul, Turkey. 

Heath, T. L. (2009). Diophantus of Alexandria: A Study in the History of Greek Algebra. Martino Fine 

Books.  

Jankvist, U. T. (2009). A characterization of the “whys” and “hows” of using history in mathematics 

education. Educational Studies in Mathematics, 71(3), 235-261. 

https://doi.org/10.1007/s10649-008-9174-9 

Katz, V. (2004). A History of Mathematics. Boston: Addison Wesley.  

Kline, M. (1990). Mathematical Thought from Ancient to Modern Times. Oxford University Press.  

Marshall, G. L., & Rich, B. S. (2000). The Role of History in a Mathematics Class. Mathematics 

Teacher, 93(8), 704-706. https://doi.org/10.5951/MT.93.8.0704 

Van der Waerden, B. L. (1983). Geometry and Algebra of Ancient Civilizations. Springer-Verlag. 

https://doi.org/10.1007/978-3-642-61779-9 

https://doi.org/10.5951/MT.86.6.0461
https://doi.org/10.1007/s10649-008-9174-9
https://doi.org/10.5951/MT.93.8.0704
https://doi.org/10.1007/978-3-642-61779-9


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Van der Waerden, B. L. (1985). “From Viète to Descartes”, A History of Algebra: From al-Khwārizmī 

to Emmy Noether. Springer-Verlag. https://doi.org/10.1007/978-3-642-51599-6_3 

Wilson, P. S., & Chauvot, J. B. (2000). Who? How? What? A Strategy for Using History to Teach 

Mathematics. Mathematics Teacher, 93(8), 642-645. https://doi.org/10.5951/MT.93.8.0642 

 

Note 

Note 1. In his paper, Réflexions sur la résolution algébrique des équations, Joseph Louis Lagrange 

introduced a new method to solve equations of low degree in a uniform way, with the hope that he 

could generalize it for higher degrees. Of course, we now know of the non-existence of an algebraic 

formula for degrees 5 and higher (Abel’s Theorem). 

 

https://doi.org/10.1007/978-3-642-51599-6_3
https://doi.org/10.5951/MT.93.8.0642

