Omar Khayyam and the Solution of Cubic Equations
Omar Khayyam and the Solution of Cubic Equations
Omar Khayyam (c. 1048-1131), also known as Umar al-Khayyam, was a Persian poet, scientist and mathematician. Khayyam's greatest work in mathematics was his enumeration of the various types of cubic equations and his solutions of each type. Khayyam's work on cubic equations was a synthesis of Greek geometry, Babylonian and Hindu arithmetic, and Islamic algebra. His work greatly influenced future Islamic mathematicians, and through them the mathematicians of Renaissance Europe.
Omar Khayyam was born in what is now Iran. He is best known in the West as a great Persian poet and philosopher. Khayyam's poetry (called the Rubaiyat of Omar Khayyam), written in the form of quatrains, was translated and adapted by Edward FitzGerald in the nineteenth century. These poems were admired as shining examples of Eastern culture. Although it is questionable just how much of the poetry attributed to Omar Khayyam was really written by Umar himself, his standing in the West has been made primarily based on his reputation as a poet.
Omar Khayyam's work as a mathematician was well known in the eastern world. However, it was not until the middle of the nineteenth century that Omar Khayyam the mathematician was "discovered" by the West, when Franz Woepke published L'algèbre d'Umar al-Khayyami. And it was not until 1931 that the mathematics of Khayyam was made available to the English-speaking world with David S. Kasir's translation of Woepke's book.
Omar Khayyam spent the majority of his life in various royal courts serving local rulers as court astronomer/astrologer. In this capacity, one of his primary duties was to create accurate calendars, an important work in medieval Islam needed to find the correct times for religious observances. It was often difficult for Khayyam to continue his work, as rulers changed and court intrigue caused him to fall out of favor with each ruler's successor.
Omar Khayyam's most important contribution to mathematics was his work involving cubic equations. A cubic equation is an equation whose highest degree variable is three, for instance x3 + 3x2 - 2x + 5 = 0. Although Khayyam did not contribute significant original methods of solution, he did write one of the first treatises that enumerated the different types of cubic equations and attempted to find the general solution of each different type of cubic equation.
In order to understand how Omar Khayyam solved cubic equations, we must first understand what mathematics was like in the eleventh and twelfth centuries. The mathematicians of the Islamic Empire were the intellectual descendants of Greek mathematicians from the last five centuries before Christ, and the Greeks had given the Islamic mathematicians geometry. Islamic mathematicians of the Middle Ages were also greatly influenced by the ancient numerical mathematics of the Babylonians and the more recent contact with the mathematics of Hindu India. The new science of algebra was being developed during this time. Even with the development of algebra, credited to al-Khwarizmi (c. 780-850), the ancient science of geometry continued to play a central role in Islamic mathematics.
In his work, On the Sphere and the Cylinder, the Greek mathematician Archimedes proposed and solved the problem, "to cut a given sphere by a plane so that the volumes of the segments are to one another in a given ratio." Since Archimedes stated that volumes of segments are related and volume is a three-dimensional measurement, this is the geometric equivalent of the algebraic solution of a cubic equation. Manaechmus, another Greek mathematician, showed how conic sections (a conic section is made by cutting a cone with a plane to produce geometric figures such as parabolas and ellipses) could be used to solve a similar geometric problem that is the equivalent to the algebra problem x3 = a2b.
The question of solving cubic equations by the use of conic sections was one that interested many Islamic mathematicians of the Middle Ages. These solutions involved a mixture of the geometric techniques inherited from the Greeks and the new algebraic techniques developed in the Islamic Empire. Mathematicians such as Ibn al-Haytham, al-Biruni, al-Mahani, and al-Khazin all contributed solutions to certain types of cubic equations. But it was Omar Khayyam who wrote the first treatise that set out the complete theory of cubic equations.
Omar Khayyam's book known as Algebra (c. 1078) contains all that is known of his work on cubic equations, as well as his work on quadratic equations. (The book's full title is Hisãb al-jabr w'al-muqãbala, which can be translated as, "The calculation of reduction and restoration." The word al-jabr in the title is an Arabic word that means "to restore", and is the origin for the word "algebra.") A quadratic equation is one whose degree is two, such as x2 + 3x - 5. However, Khayyam's work on quadratic equations was not as original as his work on cubic equations.
Omar Khayyam's work on cubic equations involved exhaustive evaluations of all the different forms of a cubic equation. For instance, he considered x3 + bx = a and x3 + a = bx to be different types of equations with distinct methods of solution. All told, Omar Khayyam provided solutions to more than a dozen different forms of cubic equations. One example of Khayyam's method for solving these equations is his solution of the equation x3 + bx = a by finding the intersection of a circle and a parabola. The intersection of these two curves is the solution of the equation. Recall that a cubic equation must have three solutions. In this case, the other two solutions are imaginary numbers, a concept that was not accepted until many centuries later; therefore, Khayyam found the only real solution to the problem. He also ignored any negative solutions that might occur.
The problems solved by Khayyam and his Islamic contemporaries were often given in terms of what we call today, "word problems." For instance, the following problem yields a cubic equation that must be solved:
Divide ten into two parts so that the sum of the squares of both parts plus the quotient obtained by dividing the greater by the smaller is equal to seventy-two.
Omar Khayyam solved the resulting cubic equation by finding the intersection of a circle and a hyperbola.
If an algebra student today were to look at the solutions of cubic equations found in Omar Khayyam's Algebra, they would likely not find much that looked familiar. Islamic algebra did not employ the symbols we use today in modern algebra. Problems and their solutions were written out in words and illustrated using geometric constructions. For instance, the problem we would write as x3 + bx = a would read as follows: "a cube and roots equal to numbers." It would actually be four centuries after Omar Khayyam before European mathematicians, particularly the Italian known as Tartaglia, developed algebraic methods and symbols similar to our modern usage.
Why then do we consider Omar Khayyam's work, and the work of many other Islamic mathematicians, as contributions to algebra? Although Omar did not use symbols and did rely on geometric constructions, his work was fundamentally algebra. This is because he sought to find solutions by assuming the solution was known (this is what we use variables for today) and proceeding to solve the problem. This procedure is exactly the same as was used by Cardano in the sixteenth century to find solutions to cubic equations using symbolic algebra much as we would use today.
Renaissance Europe owed much to these Islamic mathematicians. Translations and improvements on Greek mathematics, made by medieval Islamic mathematicians, kept alive the advanced theories and techniques that provided a foundation for Western mathematics. The West is also indebted to Islamic mathematicians for adopting and transmitting our modern number system from India. This system, which introduced the simple yet important concept of zero, has come to be called the Hindu-Arabic number system.
It is possible that Khayyam's mathematical work influenced Renaissance European mathematicians directly. Historians continue to find previously unknown connections between medieval Islamic science and scientists of Renaissance Europe. In particular, European mathematicians were influenced by an Islamic mathematician by the name of al-Tusi (sometimes spelled at-Tusi), who lived a century later than Omar Khayyam. It is known that al-Tusi was in turn influenced by the work of Omar Khayyam, although it is not clear if Renaissance Europe knew of his work.
The importance of Omar Khayyam was that he gathered together all that was known about cubic equations, as well as adding some new ideas of his own. His work formed a bridge between the strictly geometric methods of the Greeks and what would become the modern algebraic methods for solving such problems. This bridge was strengthened by Islamic mathematicians who came after Omar Khayyam. The mathematicians and scientists of Renaissance Europe were indebted to their Islamic predecessors, and these Renaissance scientists helped to form the foundation upon which the Scientific Revolution was built.
Berggren, J. L. Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag, 1986.
Kasir, D. S. The Algebra of Omar Khayyam. New York, 1931.
Katz, Victor J. A History of Mathematics. Reading, MA: Addison-Wesley, 1998.