Omar Khayyam

How to use this course

Most readers meet Omar Khayyam as the name on a Victorian book of quatrains. His contemporaries knew him as a philosopher, astronomer and geometer. His algebra gave the first systematic theory of cubic equations, his calendar was one of the most accurate ever designed, and his commentary on Euclid examined the parallel postulate with a quadrilateral that Europe rediscovered six centuries later.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proofs and exercises with folded solutions. Khayyam wrote in words, with no symbols. We translate his constructions into coordinates, and we say so each time. The biography is thin and partly legendary. Each biographical claim below names its source, and later stories are marked as such.

Timeline

Nishapur, 1048–c. 1068

Omar Khayyam was born in Nishapur, in the province of Khorasan in north-eastern Iran, in 1048. The Seljuq Turks had taken the city less than a decade earlier, and it was one of the great centres of learning in the eastern Islamic world. His name al-Khayyami means “of the tent-maker”, which may record his father Ibrahim’s trade or an earlier family one.

What the sources say

Almost everything we know comes from a handful of medieval writers, and they disagree. The most valuable is Zahir al-Din al-Bayhaqi (d. 1169–70), who met Khayyam in his youth and wrote a biographical dictionary of philosophers. Al-Bayhaqi recorded Khayyam’s horoscope: born with the Sun and Mercury in Gemini. In the twentieth century the scholar Swami Govinda Tirtha used this horoscope to compute the date 18 May 1048. The date is a reconstruction, though most modern accounts adopt it.

The second key witness is Nizami Aruzi of Samarkand, whose Chahar Maqala (Four Discourses) reports meetings with Khayyam in Balkh and Merv and a visit to his grave. The third is Ibn al-Qifti, whose thirteenth-century history of learned men is increasingly hostile in tone.

Education

Later biographical traditions name teachers in Nishapur for Quran, Arabic and the religious sciences, and a teacher who took him through Ptolemy’s Almagest. These names cannot be checked against contemporary sources. What is certain from his own works is the shape of his learning: he wrote in Arabic for scholarly subjects and in Persian for some others, and he knew Euclid, Apollonius and Archimedes, the algebra of al-Khwarizmi and his successors, and the philosophy of Ibn Sina (Avicenna), whom he regarded as his master.

A legend to set aside

A famous story makes Khayyam a schoolfellow of Nizam al-Mulk, the future vizier, and of Hasan-i Sabbah, the future leader of the Assassins. The three supposedly swore that whoever rose first would help the others. The tale appears in Persian histories written two centuries and more after Khayyam’s death, and Edward FitzGerald repeated it in 1859. Nizam al-Mulk was born about 1018, some thirty years before Khayyam, so historians treat the story as fiction.

Bukhara and Samarkand, c. 1068–1074

A scholar without inherited wealth needed a patron. Al-Bayhaqi places Khayyam at the court of the Qarakhanid ruler Shams al-Mulk Nasr in Bukhara, and around 1070 he was in Samarkand, supported by the chief judge Abu Tahir Abd al-Rahman ibn Alaq.

In the preface to his algebra Khayyam complains of the times. Scholars, he says, have been scattered or silenced, and few remain who care for real knowledge. He credits Abu Tahir with giving him the calm to write. The treatise he began in Samarkand, Risala fi’l-barahin ala masa’il al-jabr wa’l-muqabala (Treatise on the demonstration of problems of algebra), was most likely completed later, in Isfahan. It is the subject of Lessons 04 to 07.

The quadrant problem

Before the algebra Khayyam wrote a short treatise on the division of a quadrant of a circle. A geometric problem about choosing a point on a quarter circle led him to a cubic equation, which in modern form reads \(x^3+200x=20x^2+2000\). He solved it by intersecting a circle with a rectangular hyperbola and then found an approximate numerical value using trigonometric tables. In the same treatise he announced a plan: to classify all equations up to degree three and to solve each kind. The algebra carried out that plan.

Isfahan and the observatory, 1074–1092

The Seljuq sultan Malik-Shah, acting through his powerful vizier Nizam al-Mulk, summoned astronomers to build an observatory at Isfahan. The historian Ibn al-Athir, writing in the thirteenth century, lists “Umar ibn Ibrahim al-Khayyami” among them under the year 467 AH (1074–75). Khayyam remained in Isfahan for about eighteen years, the most productive period of his life.

The observatory’s tasks were practical. It produced astronomical tables, later known as the Zij-i Malikshahi, of which only a fragment survives, and it reformed the Persian solar calendar (Lesson 08). Astronomers at court were also expected to cast horoscopes. Khayyam did so, though Nizami Aruzi reports him as sceptical of predictions.

Writing in Isfahan

• 1077. The commentary on the difficult postulates of Euclid, Sharh ma ashkala min musadarat kitab Uqlidis, completed in December 1077 (Lessons 09 and 10).
• 1079. A Persian translation of a sermon by Ibn Sina, made for friends in Isfahan.
• c. 1080. Short philosophical treatises on existence and on obligation, one written for the same Abu Tahir, now a judge in Fars.

The balance

Khayyam also worked on the classical problem attributed to Archimedes: given a lump of gold alloyed with silver, find how much of each it contains without melting it. Weigh it in air and in water; the loss of weight in water is the weight of the displaced water. Al-Khazini’s Book of the Balance of Wisdom (1121) includes Khayyam’s treatment of this problem and a description of a balance he designed for it.

In 1092 this world collapsed. Nizam al-Mulk was assassinated in October, and Malik-Shah died in November. The observatory lost its funding, and Khayyam lost his protectors. Lesson 11 takes up the story.

Algebra before Khayyam and his classification

For the general reader. Algebra began in Baghdad around 820 as a method for solving quadratic equations, stated in words and justified by pictures of squares and rectangles. Khayyam asked the next question: what about cubes? He listed every possible equation of degree at most three, sorted them into types, and showed how to solve each type with geometry.

Positive coefficients, positive roots

Muhammad ibn Musa al-Khwarizmi’s algebra (c. 820) had no negative numbers and no zero coefficient. An equation such as \(x^2+21=10x\) was a different species from \(x^2=10x+21\), because each side of an equation was a sum of positive quantities. Al-Khwarizmi listed six types for degree two and solved each by completing a square drawn in the plane. The words survive in Khayyam: mal (square), jidhr or dil (root, side), ka‘b (cube), adad (number).

Later algebraists, notably al-Karaji, extended the arithmetic of polynomials. Some cubic problems had appeared too. Al-Mahani in the ninth century reduced a problem of Archimedes on cutting a sphere to a cubic but could not solve it; according to Khayyam, Abu Ja‘far al-Khazin later solved it with conic sections. Khayyam was the first to treat all cubics as one subject.

Khayyam also states that the 14 cubic types cannot, in general, be constructed with ruler and compass alone. He gave no proof, and none was possible with the tools of his time. The impossibility of duplicating the cube with ruler and compass was proved by Pierre Wantzel in 1837.

The quadratic model

Proof by picture. Draw a square of side \(x\). Attach to two adjacent sides rectangles of size \(x\times b/2\). The figure has area \(x^2+bx=d\). It lacks only a corner square of area \((b/2)^2\) to become a square of side \(x+b/2\). So \((x+b/2)^2=d+(b/2)^2\), and the result follows.

The conic toolkit

For the general reader. Greek geometers had found that some problems, like doubling the volume of a cube, could not be done with ruler and compass but could be done with curves cut from a cone. Khayyam took over these curves, mainly from Apollonius, and used them as a general machine for solving cubics.

Apollonius’ properties

Khayyam cites the Conics of Apollonius of Perga and assumes its first books. In coordinates, the two properties he uses most are:

Homogeneity

For Khayyam a number in a cubic equation is a solid, the roots are lines, and the squares are areas. An equation must compare like with like. So before solving \(x^3+cx=d\) he writes \(c\) as a square \(b^2\) and turns \(d\) into a solid on base \(b^2\): a box with base \(b^2\) and some height \(h\), so that \(d=b^2h\). This needs a lemma.

Proof. Construct the fourth proportional \(k\) with \(b:a=c:k\), so \(k=ac/b\). Then construct the fourth proportional \(h\) with \(b:a=k:h\), so \(h=ak/b=a^2c/b^2\). Each step is Euclid VI.12, done with similar triangles, and \(b^2h=a^2c\).

The cube equal to a number

Proof. Squaring the first, \(x^4=a^2y^2=a^2bx\). At the second point \(x\neq0\), so \(x^3=a^2b\). Equivalently, \(x\) and \(y\) are the two mean proportionals between \(a\) and \(b\): \(a:x=x:y=y:b\). Hippocrates of Chios had reduced doubling the cube to finding two mean proportionals, and Menaechmus found them with conics in the fourth century BC.

A cube and sides equal a number

For the general reader. This is the type \(x^3+cx=d\), the one Gerolamo Cardano would solve by a formula four and a half centuries later. Khayyam found its root as the height of the point where a parabola crosses a circle. The picture at the top of this page is his construction.

Proof

At an intersection point, the parabola gives \(x=y^2/b\). Substitute into the circle:

Khayyam argues with proportions instead. From the parabola, \(b:y=y:x\). From the circle, \(y:x=x:(h-y)\). So \(b,\ y,\ x,\ h-y\) are in continued proportion, and therefore \(b^2:y^2=y:(h-y)\). Cross-multiplying gives the cube \(y^3\) equal to the solid \(b^2(h-y)\), and adding \(b^2y\) to both sides gives the result.

Existence and uniqueness. The function \(f(y)=y^3+cy\) is strictly increasing on \(y>0\), with \(f(0)=0\) and \(f(y)\to\infty\), so it takes the value \(d\) exactly once. Geometrically, the circle lies in the strip \(0\le y\le h\), is tangent to the \(x\)-axis at \(B\), and the parabola leaves \(B\) tangent to the \(y\)-axis, so the parabola starts inside the circle and must cross it once before \(y\) reaches \(h\).

Circles, hyperbolas and the number of roots

For the general reader. Some cubic types can have no positive solution, or two. Khayyam saw this: in his constructions the two curves may miss each other, touch, or cross twice. He did not notice that some types can have three positive solutions. This lesson works through the quadrant equation of Lesson 02 and the question of how many roots there are.

A cube and sides equal squares and a number

Khayyam handles the type \(x^3+cx=ax^2+d\), to which the quadrant equation belongs, by a circle and a rectangular hyperbola. The following coordinate version follows his idea. Write \(c=k^2\) and \(m=d/k^2\). The equation becomes

Proof. On \(\mathcal H\), \(x^2y^2=k^2(m-x)^2\). On \(\Gamma\), \(y^2=(x-a)(m-x)\). Combining, \(x^2(x-a)(m-x)=k^2(m-x)^2\). Divide by \(m-x\). Conversely, a root \(x\neq m\) gives \(y=k(m-x)/x\), and the equation shows that this \(y\) satisfies \(y^2=(x-a)(m-x)\). The hyperbola is rectangular: it is \(x(y+k)=km\), with asymptotes \(x=0\) and \(y=-k\).

Any root must lie between \(a\) and \(m\), since \(x^2(x-a)\) and \(k^2(m-x)\) must have the same sign. Khayyam discusses such conditions in words: if the number is too large or too small relative to the other terms, the curves do not meet and the problem is impossible.

How many roots?

For several types Khayyam notes that the curves may meet in two points, giving two solutions, or touch, giving one, or miss, giving none. An example is \(x^3+d=cx\). The type \(x^3+cx=ax^2+d\), however, can have three positive roots, and Khayyam does not say so. Recognizing all real roots of cubics, and the role of negative and complex ones, waited for the sixteenth-century Italians and for René Descartes’ rule of signs (1637).

The Jalali calendar

For the general reader. A calendar must keep the seasons fixed, so the length of its average year must match the time the Sun takes to return to the same equinox. That interval is not a whole number of days. The Isfahan astronomers designed a calendar for Malik-Shah that tracked the spring equinox more closely than the Gregorian calendar adopted in Europe five centuries later.

What is documented

The old Persian solar year had 365 days with no leap day, so New Year (Nowruz) wandered through the seasons. Malik-Shah ordered a reform, carried out by a group of astronomers at Isfahan of whom Khayyam was one. The new era, named Jalali after the sultan’s title Jalal al-Dawla, was inaugurated in March 1079 (10 Ramadan 471 AH; modern conversions give 15 or 16 March). The year began at the spring equinox, and months were tied to the Sun’s passage through the signs of the zodiac. Khayyam is credited with a value of the tropical year of 365.24219858156 days, an extraordinary number of digits for an observed quantity; the modern mean tropical year is about 365.24219 days.

What is uncertain

As designed, the calendar placed Nowruz by astronomical computation of the equinox, so it needed no fixed leap-year rule. Later writers attributed to Khayyam an arithmetic rule of 8 leap years in every 33, and other cycles have been proposed. The surviving sources do not settle which rule, if any, the astronomers fixed. The calendar was used in Iran in various forms for centuries; in 1925 it was simplified into the present Solar Hijri calendar.

The Julian calendar uses \(1/4\). The 33-year rule uses \(8/33\). The Gregorian calendar uses \(97/400=0.2425\), which is not a convergent and is less accurate than \(8/33\). One subtlety: a calendar that keeps the spring equinox fixed should match the vernal equinox year, currently about 365.2424 days, which is slightly longer than the mean tropical year. The 33-year cycle fits that value even better.

The parallel postulate

For the general reader. Euclid’s fifth postulate, about when two lines meet, always seemed less obvious than the others. For two thousand years geometers tried to prove it from the rest. Khayyam’s attempt, written in Isfahan in 1077, introduced a four-sided figure that became the standard tool for the problem and examined the three possibilities for its angles. Those three possibilities correspond to the three geometries known today.

Khayyam’s critique

Khayyam’s commentary opens by criticizing earlier attempts, including that of Ibn al-Haytham, for using motion in geometry: Ibn al-Haytham had a line segment slide along another, which Khayyam judged foreign to geometric proof. Khayyam then replaces the postulate with a principle he attributes to “the philosopher”, meaning Aristotle:

The quadrilateral

Take a segment \(AB\). Erect equal perpendiculars \(AC\) and \(BD\) on the same side, and join \(CD\). This is the figure now called the Khayyam–Saccheri quadrilateral, after Giovanni Girolamo Saccheri, who used it in 1733. Khayyam proves, without the fifth postulate:

The summit angles are therefore both right, both obtuse or both acute. Khayyam rejects the obtuse and acute cases with his principle. In the obtuse case, he argues, the perpendiculars \(AC\) and \(BD\) would converge as one moves up from the base. Repeating the figure along the other side of the base would make them converge in the other direction too, so the lines would be diverging in the direction of convergence, which the principle forbids. The acute case is excluded symmetrically. From right summit angles he goes on to derive Euclid’s postulate.

Assessment

The principle is equivalent to the fifth postulate given the other axioms, so the argument replaces one assumption with another. The value of the work lies elsewhere. Khayyam isolated the three hypotheses on the summit angles and proved real theorems in absolute geometry. Nasir al-Din al-Tusi discussed Khayyam’s argument in 1250. A later work in al-Tusi’s school, printed in Rome in 1594, was read in Europe, and Saccheri’s book opens with a criticism of John Wallis, who had studied that text. Saccheri’s acute-angle hypothesis turned out to describe hyperbolic geometry, found by János Bolyai and Nikolai Lobachevsky around 1830. Whether Saccheri knew Khayyam’s work is not established.

Ratio, number and roots

For the general reader. Greek mathematics distinguished numbers from magnitudes such as lengths. A ratio of two lengths, like the diagonal of a square to its side, need not be a ratio of whole numbers. The second and third books of Khayyam’s Euclid commentary examine what it means for two ratios to be equal, and ask whether a ratio might itself be a kind of number.

Euclid used the first in Book V; Arabic scholars, including al-Mahani, preferred the second, which they found closer to the meaning of ratio. Khayyam proved that the two definitions agree. In modern terms, the quotient sequence is the continued fraction of \(a/b\), and two positive reals have the same continued fraction exactly when they are equal, which is also what the Eudoxus test detects.

In Book III Khayyam treats the composition of ratios, \((a:b)\cdot(b:c)=a:c\). To handle it, he chooses a unit and considers, for any ratio, a magnitude standing to the unit in that ratio. He then asks whether such a magnitude should be counted among the numbers. He leaves the question open, but the step points toward treating any ratio, rational or not, as a number.

Roots and binomial coefficients

In the algebra Khayyam mentions a separate work, now lost, in which he extended Indian methods of extracting square and cube roots to roots of any order. Historians infer that such a method rests on the expansion of \((a+b)^n\) and its coefficients, the triangle known in Europe as Pascal’s. Al-Karaji had described that triangle earlier. Because the work is lost, Khayyam’s exact method is unknown.

After 1092: Merv and Nishapur

The deaths of Nizam al-Mulk and Malik-Shah in late 1092 opened years of struggle among the sultan’s heirs. Khayyam lost his position at court. The sources for the next forty years are scattered and sometimes contradictory.

• Pilgrimage. Ibn al-Qifti, writing more than a century later, says Khayyam made the pilgrimage to Mecca and stayed in Baghdad. He presents it as a show of piety to silence critics, a reading coloured by his own hostility. The pilgrimage itself is commonly accepted; its motive is not known.
• Sanjar. Al-Bayhaqi tells that Khayyam treated the young prince Sanjar during smallpox and gave a gloomy prognosis, and that Sanjar held this against him. Sanjar governed Khorasan from 1097 and became supreme sultan in 1118, with his capital at Merv.
• Balkh and Merv. Nizami Aruzi met Khayyam in Balkh in 1112–13, in the company of Abu Hatim al-Isfizari, and again in Merv in the winter of 1114–15, where both were guests of the vizier. Al-Isfizari and Khayyam had worked together; al-Isfizari’s own studies of the balance appear in al-Khazini’s book.
• Debate. The grammarian al-Zamakhshari describes a debate with Khayyam in Khwarazm and praises him while claiming victory.

Death

Khayyam died in Nishapur. The date 4 December 1131 is widely given; older literature offers c. 1123, and the medieval sources do not fix it. Nizami Aruzi wrote that he visited the grave in Nishapur in 1135–36 and found it beside a garden wall, under fruit trees whose blossom covered the stone, just as Khayyam had once predicted to him. The story is charming and impossible to check.

Afterlives

In mathematics

Khayyam’s geometric theory of cubics was taken further by Sharaf al-Din al-Tusi (d. c. 1213), who studied the existence and number of positive roots through the maximum of a cubic expression. In Europe the algebraic solution came from Scipione del Ferro, Niccolò Tartaglia and Gerolamo Cardano in the 1500s, without knowledge of Khayyam. René Descartes, in La Géométrie (1637), again solved cubics and quartics by intersecting a parabola with a circle, an approach close to Khayyam’s. The algebra became known in Europe only when Franz Woepcke edited and translated it in 1851.

The quadrilateral of the Euclid commentary passed to Nasir al-Din al-Tusi and his school. Giovanni Girolamo Saccheri’s Euclides ab omni naevo vindicatus (1733) used the same figure and the same three hypotheses, and Johann Heinrich Lambert pushed the acute case further. The acute hypothesis is consistent: it is hyperbolic geometry.

The calendar

The Jalali reform shaped Persian timekeeping for nine centuries. Iran adopted a solar calendar officially in 1911 and simplified it in 1925 into the Solar Hijri calendar still used in Iran and Afghanistan, with the year beginning at the spring equinox.

The quatrains

Persian quatrains (ruba‘iyat) were attributed to Khayyam from the late twelfth century, and manuscripts grew larger with time; a Bodleian manuscript copied in Shiraz in 1460 contains 158. Which, if any, are his is disputed, and some scholars doubt the attribution altogether. Edward FitzGerald’s free English version of 1859 sold slowly at first and then became one of the most popular poems in the language, making Khayyam’s name famous in the West for reasons that have little to do with his science.

Honours

• A mausoleum designed by Houshang Seyhoun stands over his tomb in Nishapur.
• The lunar crater Omar Khayyam (named 1970) and the asteroid 3095 Omarkhayyam (discovered 1980).
• Iran donated a pavilion in his honour to the United Nations Office in Vienna in 2009.

Connections

People linked to Omar Khayyam across this collection. A name without a link has no page here yet. See Omar Khayyam in the network of all 337 lives →

Influenced by

• Apollonius of Perga
• Archimedes
• Euclid
• Ibn al-Haytham
• Ibn Sina
• Muhammad ibn Musa al-Khwarizmi
• al-Karaji
• al-Mahani

Influenced

• Fibonacci
• Giovanni Girolamo Saccheri
• Nasir al-Din al-Tusi
• René Descartes
• al-Khazini
• Sharaf al-Din al-Tusi

Colleagues and collaborators

• Abu Hatim al-Isfizari 1074-1115

Also mentioned in this life

• Aristotle
• János Bolyai
• Gerolamo Cardano
• Eudoxus of Cnidus
• Johann Heinrich Lambert
• Nikolai Lobachevsky
• Blaise Pascal
• Claudius Ptolemy
• Niccolò Tartaglia
• John Wallis

Fields

• algebra 29 lives
• geometry 32 lives
• astronomy 50 lives
• philosophy 12 lives

On CoursesHub

• Astrophysics for Beginners

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Omar Khayyam, MacTutor History of Mathematics.
2. Khayyam, Omar: Life, Encyclopaedia Iranica (the medieval sources discussed one by one).
3. Omar Khayyam, Wikipedia.
4. Umar al-Khayyam, Muslim Heritage (works and dates).
5. Nizami Aruzi, Chahar Maqala, trans. E. G. Browne, 1921.

Mathematics

1. F. Woepcke, L’algèbre d’Omar Alkhayyâmî, Paris, 1851.
2. D. S. Kasir, The Algebra of Omar Khayyam, Columbia University, 1931.
3. R. Rashed and B. Vahabzadeh, Omar Khayyam the Mathematician, Bibliotheca Persica, 2000.
4. J. L. Berggren, Episodes in the Mathematics of Medieval Islam, Springer, 2nd ed. 2016, chapter on algebra.
5. Saccheri quadrilateral and Parallel postulate, Wikipedia (history sections).
6. Jalali calendar, Wikipedia.
7. B. A. Rosenfeld, A History of Non-Euclidean Geometry, Springer, 1988.
8. Note sur le choix des courbes fait par al-Khayyam, arXiv:0912.1418 (Khayyam’s curves compared with Descartes’).