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Omar Khayyam's Al-jabr considered six types of cubic equations in which one term was set equal to a sum of two others. Use the following interactive sliders to explore the geometric solution of the cubic equation

3

x

+axb

involving a semicircle and a parabola, as given by Khayyam:

Out[]=

​
a b

Computational Explanation

Omar Khayyam was a Persian poet, philosopher, mathematician and astronomer who lived in the second half of the eleventh century. In the Western world, he is best known for the nearly six hundred short poems appearing in the Rubáiyát. Khayyam also authored the treatise On the Proofs of the Problems of Algebra and Muqabala, commonly known as Khayyam's Al-jabr, around 1070–1079 CE. In this work, Khayyam used mainly geometric methods to solve cubic equations.

While Khayyam's work as a mathematician was widely known in the Eastern world, his mathematical work was not known in the west until 1851 when Franz Woepke published a French translation of his work. The first English treatment of Khayyam's mathematics did not appear until even later, when David S. Kasir published The Algebra of Omar Khayyam in 1932.

Khayyam used geometric methods combined with the notion of proportions of geometric lengths to solve these equations. The sorts of geometric constructions he considered are exemplified by the following diagram from his Al-jabr:

Khayyam considered six types of cubic equations in which one term (a monomial) was set equal to a sum of two other terms (a binomial):

◼

3

x

+axb

◼

3

x

+bax

◼

3

x

ax+b

◼

3

x

+a

2

x

b

◼

3

x

+ba

2

x

◼

3

x

a

2

x

+bx

To solve the first cubic equation

3

x

+axb

, written by Khayyam as "A cube and (a number of) sides are equal to a number," Khayyam first rewrote the original equation in homogenous form:

3

x

+

2

c

x+

2

c

h,

… with the constants c and h given in terms of the original variables a and b as:

In[]:=

Solve[{a

2

c

,

2

c

hb},{c,h}]//Last

Out[]=

c

a

,h

b

a



Khayyam then constructed the following figure:

Out[]=

Here, the blue curve is a downward-opening parabola with vertex at B and latus rectum length

c

AB

, and CB is a diameter of a semicircle with

CB

h

. These two curves intersect in a unique point D. Let

xDZ

and

yDE

be the coordinates of this point of intersection regarded as variables. The semicircle has the equation:

2

y

x(h-x),

… which Khayyam described in terms of the proportion "BE is to ED as ED is to EC":

x

y



y

h-x

.

Similarly, the parabola has the equation:

yc

2

x

,

… which Khayyam described in terms of the proportion "AB is to BE as BE is to ED":

c

x



x

y

.

From these two proportions, Khayyam correctly concluded that

EBx

is a solution to the original equation, and in fact the only solution. The validity of this solution can immediately be verified by eliminating

y

from the simultaneous equations after substituting the original variables back in:

In[]:=

Assuminga≠0&&x≠0,FullSimplifyEliminate{yc

2

x

,

2

y

x(h-x)}/.c

a

,hba,y//Expand

Out[]=

bax+

3

x

Khayyam solved the other cubic equations using similar methods of finding the intersections of conic sections.

Although similar methods had been used by Menaechmus to solve a problem equivalent to the cubic equation

3

x



2

a

b

as early as around 350 BCE and (as noted by Khayyam) by tenth-century Persian mathematician Abū al-Jūd for a particle cubic equation, Khayyam's work in Al-jabr represents the first systematic study of—as well as the first exact solution procedure for—general classes of cubic equations.

Additional Reading

• Berggren, J. L. Episodes in the Mathematics of Medieval Islam. New York: Springer-Verlag, 1986.
• Berggren, J. L. "The Algebra of Omar Khayyām." In The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 556–560, 2007.
• Grattan-Guinness, I. (Ed.) Companion Encyclopedia on the History and Philosophy of the Mathematical Science, Vol. 1. London: Routledge, p. 76, 1994.
• Kasir, D. S. The Algebra of Omar Khayyam. New York: Columbia University, 1932.
• Katz, V. J. A History of Mathematics. Reading, MA: Addison-Wesley, 1998.
• Mardia, K. V. "Omar Khayyam, René Descartes and Solutions to Algebraic Equations." Paper presented to the Omar Khayyam Club, London, 19 November 1999. [link]
• Merzbach, U. C. and Boyer, C. B. "Omar Khayyam." In A History of Mathematics, 3rd ed. Hoboken, NJ: Wiley, pp. 218–220, 2011.
• van der Waerden, B. L. "Omar Khayyam." Ch. 1, Part C in A History of Algebra: From al-Khwārizmī to Emmy Noether. Berlin: Springer-Verlag, pp. 24–31, 1985.
• Woepke. F. L'algèbre d'Omar Alkhayyami. Paris: Benjamin Duprat, 1851.

Additional Links

• Columbia University Libraries: Khayyam's Maqalah fi al-jabr wa-al muqabalah
• Encyclopedia.com: Omar Khayyam And The Solution Of Cubic Equations
• Jewels in Her Crown: Treasures of Columbia University Libraries Special Collections: Maqalah fi al-jabr wa-al muqabalah
• Jewels in Her Crown: Treasures of Columbia University Libraries Special Collections: 159. Omar Khayyam
• MacTutor History of Mathematics Website: Omar Khayyam
• Mathematical Treasures - Omar Khayyam's Algebra
• Wikipedia: Omar Khayyam
• Wikipedia: Cubic equation: Omar Khayyám's solution