Astronomy and Mathematics under the Seljuks

In eleventh-century Iran, mathematics and astronomy served both scholarly inquiry and court needs. Omar Khayyam studied equations and Euclid’s geometry, while Sultan Malik Shah and his chief minister Nizam al-Mulk sponsored astronomers to align the new year with the spring equinox. Mathematics advanced through geometric proof; the calendar project depended on watching the sun’s annual course.

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Khayyam and cubic equations

A cubic equation includes an unknown multiplied by itself twice, making three equal factors. For example, if a cube has sides three units long, its volume is three times three times three. Khayyam described an unknown as a “thing,” a line length; its square was an area and its cube a solid volume. He classified cubic problems by how these positive magnitudes, squares, cubes and known numbers were related. This way of treating the unknown as a measurable length shaped how he built solutions.

For difficult cubic forms, Khayyam used conic sections, curves made by cutting a cone with a plane. A circle is closed around its center; a parabola is an open U-shaped curve; a hyperbola has two separate branches. Each curve represented a geometric condition in the problem. Where two curves crossed, both conditions held at once, and the intersection let a geometer determine the unknown length.

A modern reconstruction of one of Khayyam’s solutions shows how this worked. For a cubic in which a cube and a multiple of the unknown length equal a known quantity, the construction pairs a parabola with a semicircle. At their intersection, the construction drops a line at a right angle to the base. The segment of the base from its starting point to where the perpendicular meets it gives the unknown length. Khayyam used different pairs of conics for different cubic forms, organizing a family of problems into methods that could be demonstrated geometrically.

Euclid, parallels and proportion

Khayyam also studied Euclid’s Elements, a foundational Greek work on geometry. He examined the parallel postulate, a rule about when two straight lines meet after a third line crosses them. To test it, he considered a four-sided figure whose two equal sides rose at right angles from its base. He then examined whether its other angles could be right, acute (smaller than a right angle) or obtuse (larger than a right angle). By comparing the possible angles, he tried to establish the postulate rather than simply accept it.

Another problem concerned ratios. A ratio compares magnitudes, such as the lengths of two lines. Khayyam examined definitions of equal ratios and sought to reconcile Euclid’s approach with arguments developed by earlier mathematicians working in Arabic. In these treatises he treated lines, areas and solids as continuous magnitudes, which let him connect numerical relationships to geometric constructions.

Observing the spring equinox

Malik Shah’s astronomers addressed a mismatch between calendar years and the seasons. A lunar calendar counts months by the moon; twelve lunar months make a year shorter than the sun’s seasonal cycle, so the dates move through the seasons. The Persian administrative year also used a fixed 365-day count, so its beginning could slip against the agricultural year. These mismatches affected annual obligations and the placement of Nowruz, the traditional Persian new year.

The seasonal year is a little longer than 365 days, so a calendar kept at exactly 365 days gradually moves away from spring. Astronomers observed the sun’s course to locate the spring equinox. The sun’s apparent yearly path crosses the celestial equator, an imaginary line extending Earth’s equator into the sky; the spring crossing marks the turn into spring in the Northern Hemisphere. Observations of that crossing gave timekeepers a reference for setting the new year.

Accounts describe Malik Shah and Nizam al-Mulk gathering Khayyam and other astronomers, including Abu’l-Muzaffar al-Isfizari and Maymun ibn al-Najib al-Wasiti. Sources associate the observations with Isfahan, though they differ about the project’s exact arrangements and location. Their task required comparison of the sun’s position across the year so the spring marker could be identified.

The Jalali calendar

The calendar reform became known as the Jalali calendar, after Malik Shah’s honorific Jalal al-Din. Its era began in March 1079, with the new year set at the spring equinox. This solar reckoning kept annual dates tied to the seasons, while the Islamic lunar calendar continued to set religious months. Ramadan, for example, remained tied to the lunar calendar and therefore moved through the solar seasons.

Accounts report that observations continued until Malik Shah’s death in 1092, when the court-sponsored project was abandoned. Khayyam’s mathematical writings continued through copying and study beyond the observatory project. Their later readers depended on manuscript transmission as well as on scholarly traditions that preserved and discussed earlier work.