"What remains from among the most important and difficult problems [to solve] is the difference among the order of existents...
Perhaps I, and my teacher, the master of all who have proceeded before him, Avicenna(Ibn-i Sina), have thoughtfully reflected
upon this problem and to the extent that it is satisfactory to our intellects, we have understood it."
Omar Khayyam, Arabic in full Ghiyath al-Din Abu al-Fath Umar ibn Ibrahim al-Nisaburi al-Khayyami (born May 18, 1048, Neyshabur , Khorasan [now Iran]—died December 4, 1131, Neyshabur), Persian mathematician, astronomer, and poet, renowned in his own country and time for his scientific achievements but chiefly known to English-speaking readers through the translation of a collection of his roba'iyat (“quatrains”) in The Rubáiyát of Omar Khayyám (1859), by the English writer Edward FitzGerald.
Khayyam received a good education in the sciences and philosophy in his native Neyshabur before traveling to Samarkand,
where he completed the algebra treatise, Risalah fi'l-barahin ala masa'il al-jabr wa'l-muqabalah (“Treatise on Demonstration of Problems of Algebra”), on which his mathematical reputation principally rests. In this treatise he gave a systematic discussion of the solution of cubic equations by means of intersecting conic sections. Perhaps it was in the context of this work that he discovered how to extend Abu al-Wafa’s results on the extraction of cube and fourth roots to the extraction of nth roots of numbers for arbitrary whole numbers n.
During this time, Khayyam led work on compiling astronomical tables, and he also contributed to calendar reform; in 1079 Khayyam measured the length of the year as 365.24219858156 days.
Khayyam is an early work on algebra written before his famous algebra text. In it he considers the problem:
Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii,
the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal.
Khayyam shows that this problem is equivalent to solving a second problem:
Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse.This problem in turn led Khayyam to solve the cubic equation and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by ruler and compass methods, a result which would not be proved for another 750 years.
Quadrilateral of Omar Khayyam
Omar Khayyam's Algebra
" The Algebra of Omar Khayyam" by Daoud S. Kasýr. Teachers College,
Columbia University Contributions to Education, No: 385