Omar Khayyam Solves Cubic Equations With Conic Sections
A Persian poet and mathematician classifies every type of cubic equation and solves them by intersecting curves, five centuries before algebra catches up
Quick facts
- Khayyam's dates
- 1048-1131
- Key mathematical achievement
- Geometric classification and solution of cubic equations via conic sections
- Calendar reform
- 1079, Jalali calendar
- Literary fame
- The Rubaiyat (Fitzgerald translation, 1859)
What happened
Omar Khayyam, born in 1048 in Nishapur, Persia, is remembered today largely for the Rubaiyat, the collection of nearly 600 four-line verses made famous in the West by Edward Fitzgerald's 1859 translation, but his mathematical work was substantial in its own right. In a treatise on algebra, Khayyam gave a complete classification of cubic equations and, for each type, a geometric method of solving it by intersecting conic sections, such as a parabola with a circle, results Greek mathematicians had never achieved for cubics despite mastering conics themselves. He stated his ambition directly: if the opportunity arose he intended to give all fourteen forms of cubic equations with their branches and cases, and how to distinguish whatever is possible or impossible in each. Separately, in 1079, Khayyam was one of eight scholars commissioned to reform the Persian calendar, producing the Jalali calendar based on a year length of 365.24219858156 days, a figure very close to the modern measurement.
Why it matters
Khayyam's geometric classification of cubic equations extended what the Greeks had done with conic sections into territory Greek mathematicians never reached, and his work, along with related efforts by other Islamic mathematicians such as Sharaf al-Din al-Tusi, fed directly into the algebraic tradition that Italian mathematicians would draw on when they finally solved the cubic algebraically in the 16th century.
How we know
Khayyam's algebra treatise survives in Arabic manuscript and has been translated and studied by historians of mathematics, and his calendar reform work is independently corroborated by the accuracy of the Jalali calendar's year length against modern astronomical measurement.
Sources
- MacTutor History of Mathematics, University of St Andrews. Omar Khayyam · Reputable sourcemathshistory.st-andrews.ac.uk · The domain "mathshistory.st-andrews.ac.uk" is on our Reputable source registry. · Link is live and its text matches the event's key terms (Jul 2026)
- MacTutor History of Mathematics, University of St Andrews. Arabic mathematics · Reputable sourcemathshistory.st-andrews.ac.uk · The domain "mathshistory.st-andrews.ac.uk" is on our Reputable source registry. · Link is live and its text matches the event's key terms (Jul 2026)
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Related timelines
- The Rise of Islam → · See the Rise of Islam timeline for the wider Persian and Islamic intellectual world Khayyam worked in during the Seljuk period.