Omar Khayyām

Omar Khayyām was born on 18 May 1048 in Nishapur, in the Khorāsān region of Persia (modern Iran), and died there on 4 December 1131. The name al-Khayyāmī translates literally as tent maker. In the West he is chiefly known through Edward FitzGerald's 1859 loose translation of some 600 quatrains, the Rubaiyat, though scholars can attribute only about 120 of those verses to him with confidence. In the East, and to a mathematics classroom, his primary significance is the algebra treatise.

His life was framed by the Seljuq turbulence of the eleventh century. He studied at Nishapur, moved to Samarkand around 1070 under the patronage of the jurist Abu Tahir, and there completed the Treatise on Demonstration of Problems of Algebra. A subsequent invitation from Sultan Malik-Shāh and his vizier Niẓām al-Mulk brought him to Isfahan to run a new observatory, where he worked for eighteen productive years. When Niẓām al-Mulk was assassinated in October 1092 and Malik-Shāh died a month later, the observatory's funding ended, and Khayyām spent the remaining decades in comparative disfavour, eventually moving to Merv (modern Mary, Turkmenistan) under Sultan Sanjar.

Principal works

• Risālah fī al-barāhīn ʿalā masāʾil al-jabr wa-l-muqābalah (Treatise on Demonstration of Problems of Algebra), the algebra treatise on which his mathematical reputation principally rests. Britannica gives the full Arabic title.
• Sharḥ mā ashkala min muṣādarāt kitāb Uqlīdis (Commentaries on the Difficulties in the Postulates of Euclid), his critical treatment of the parallel postulate and of the Euclidean theory of ratio.
• Problems of Arithmetic, a book on music, and an earlier work on algebra, all written before he was twenty five and referred to in his later writings; the earlier algebra survives only through reference in the mature Treatise.
• A lost work, referred to in his algebra, in which he discusses the binomial coefficients arranged in the array now called Pascal's triangle.
• The Rubaiyat, the poetry collection through which he became famous in Europe after FitzGerald's 1859 translation.

Cubic equations: the classification

The centrepiece of Khayyām's algebra is a systematic classification of cubic equations, all expressed with positive coefficients. Two numbers appear in the literature, and both are correct because they refer to different things.

Twenty five species is the count when every equation of degree at most three is listed in Khayyām's own terms, a cube plus sides equal numbers, a cube equal squares plus numbers, and so on. Roshdi Rashed's critical edition uses this count; Katz's History of Mathematics gives the same figure, describing the twenty five as 11 by Euclidean methods and 14 by conic sections. The Stanford Encyclopedia of Philosophy expresses the essential mathematical point in a way that a modern reader may find clearer: fourteen distinct types of equations cannot be reduced to a linear or quadratic form after dividing by x or x².

Fourteen forms is Khayyām's own phrase, quoted in MacTutor: If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible. These fourteen are the irreducible cubics, the forms that remain cubic after division or substitution. For each one, he shows how the intersection of two suitably chosen conic sections produces the required root as the length of a line segment.

He states that this class of equations requires conic sections beyond ruler and compass methods, a claim it took another 750 years to prove rigorously (in the nineteenth century, via Wantzel and Galois theory). He is careful to observe that a cubic may have more than one positive solution, and demonstrates the existence of equations with two positive roots. MacTutor preserves an honest limit: he does not appear to have found that a cubic can have three solutions.

The nth root and the binomial coefficients

In a passage MacTutor quotes at length, Khayyām writes that the Indians possess methods for finding the sides of squares and cubes based on such knowledge of the squares of nine figures, and reports that he has extended these methods to find the sides of the square square, quatro cube, cubo cube, etc. to any length, which has not been made before now. The technique amounts to the binomial expansion, and in a companion, now lost, he discusses the coefficient array now known as Pascal's triangle. The extraction of general nth roots is, in this framing, an original contribution of his own; Stanford summarises Rashed and Vahabzadeh (2000): he is the first to extend these methods to the determination of roots of any order.

The parallel postulate

In the Commentaries on the Difficulties in the Postulates of Euclid, Khayyām attempted to prove Euclid's fifth postulate from the other four. MacTutor is careful about the interpretation: In trying to prove the parallels postulate he accidentally proved properties of figures in non Euclidean geometries, and further, this was not his intention. The constructions he introduced, particularly the analysis of a quadrilateral now known in the West as the Saccheri quadrilateral, would, six centuries later, feed directly into the emergence of hyperbolic geometry.

The same treatise contains substantive work on Euclidean ratio: he proved that Euclid's definition of equality of ratios and the definition proposed by earlier Islamic mathematicians (such as al-Māhānī) are equivalent, extended the Euclidean theory to include the multiplication of ratios, and raised the open question of whether a ratio may be regarded as a number. Britannica notes that Khayyām's ideas on the parallel postulate eventually made their way to Europe, where they influenced John Wallis (1616 to 1703).

The Jalālī calendar

Malik-Shāh brought together eight scholars, Khayyām among them, to reform the calendar. The result, the Jalālī calendar, was adopted in 1075 and organised leap years on a 33 year cycle, eight leap years in every thirty three. Britannica notes it was more accurate than the present Gregorian calendar. MacTutor records Khayyām's measured year length as 365.24219858156 days, a figure preserved in the biography to full decimal precision, though the number of significant places substantially exceeds what an observational programme of that era could have determined; we cite the figure as MacTutor gives it, and note that the calendar system's leap year cycle is the more historically verifiable achievement.

Related curriculum links in the curriculum

Khayyām's algebra follows Al-Khwārizmī's treatment of the quadratic and extends the geometric solution programme to the cubic. His work on ratio and on the parallel postulate provides one of the historical precedents for later work in non Euclidean geometry, the concerns that motivate the axiomatic style of much modern olympiad geometry. The Continuum places him in a shorter chronological arc; the About page's Heritage section gives the wider institutional framing.

Sources

1. MacTutor History of Mathematics, University of St Andrews, Omar Khayyām biography (birth 18 May 1048 in Nishapur, death 4 December 1131 in Nishapur, full name including Ghiyāth al-Dīn, tent maker etymology, the Samarkand period under Abu Tahir, the Isfahan observatory under Malik-Shāh, the 1092 political rupture, the four titled works, the fourteen forms direct quote, the ruler and compass observation and its 750 years later resolution, the hedge that Khayyām does not appear to have found that a cubic can have three solutions, the nth root passage from the Indian methods, the Pascal triangle reference, the parallel postulate work with the this was not his intention hedge, and the calendar work). https://mathshistory.st-andrews.ac.uk/Biographies/Khayyam/
2. Encyclopædia Britannica, Omar Khayyam biography (verification of dates 18 May 1048 to 4 December 1131, Neyshābūr, the full Arabic title of the algebra treatise, the general framing of cubic equation classification and geometric solution, the influence on Wallis, and the Jalālī calendar's 8 in 33 leap year rule adopted in 1075 by Malik-Shāh). https://www.britannica.com/biography/Omar-Khayyam-Persian-poet-and-astronomer
3. Stanford Encyclopedia of Philosophy, Umar Khayyam entry (the fourteen distinct types of equations cannot be reduced to a linear or quadratic form after dividing by x or x² characterisation, and the Rashed & Vahabzadeh 2000 attribution of the extension of nth root methods). https://plato.stanford.edu/entries/umar-khayyam/
4. Katz, Episodes in the Mathematics of Medieval Islam, chapter on al-Khayyāmī (the twenty five species count with the split into 11 by Euclidean methods and 14 by conic sections). https://www.ms.uky.edu/~dhje223/EpisodesInTheMathematicsOfMedi-pages-157-161.pdf
5. Aghayani-Chavoshi & Bagheri (Suhayl), Al-Khayyām's Scientific Revision of Algebra (the twenty five species count and the reason positive coefficient convention produces that number). https://www.ub.edu/arab/suhayl/volums/volum10/2_paper2Vol10.pdf