Astronomy and Mathematics in Timurid Samarkand

In fifteenth-century Samarkand, Ulugh Beg supported scholars who taught mathematics, constructed instruments and made astronomical observations. The programme brought together Qadi Zada al-Rumi, al-Kashi and Ali Qushji. Its principal outcome was a set of tables associated with Ulugh Beg, commonly called the Zij-i Sultani.

On this page5 sections

Teaching and the observatory

Ulugh Beg’s madrasa provided a setting for advanced teaching before the observatory was completed. Al-Kashi’s letters to his father describe discussions, scholarly rivalry and Ulugh Beg’s personal interest in mathematical problems. They record a patron taking part in learned work as well as paying for it.

The observatory was developed during the 1420s. Al-Kashi worked on its instruments and calculations, but died before the final tables were completed. Qadi Zada and Ali Qushji participated in the continuing programme. Its work extended across years and depended on several scholars.

Measuring with a large meridian arc

The Ulugh Beg Observatory preserves part of a huge curved instrument sunk into the ground. Its surviving parallel masonry arcs belonged to a meridian instrument, aligned north–south to measure celestial altitudes as bodies crossed that line. The full structure above ground has disappeared; the present museum and shelter are modern.

A large radius spread a small angle across a longer physical distance, making its divisions easier to read. Accurate alignment and stable construction remained essential. Al-Kashi’s letters discuss the practical problem of making instruments large enough for precise measurement without allowing their materials to warp.

The star catalogue and working tables

A zij supplies numerical tables and instructions for astronomical calculations. Ulugh Beg’s tables included a catalogue of more than a thousand stars, alongside material used to calculate celestial positions and calendars. Much of the catalogue was based on new observation; stars too far south to observe from Samarkand required earlier information.

The catalogue gave positions relative to the celestial coordinate system. A later user needed to account for the gradual change in those coordinates over time. The tables thus preserved the results of observations at a particular epoch, rather than a timeless picture of the sky.

Al-Kashi’s calculations

Al-Kashi worked with sexagesimal notation, based on sixty, and decimal fractions. Degrees divided into minutes and seconds made sexagesimal arithmetic especially convenient for astronomy. His Key to Arithmetic organised methods of calculation, while separate works treated geometrical and trigonometric problems.

To approximate the ratio of a circle’s circumference to its diameter, he calculated with polygons drawn inside and outside a circle. Increasing the number of sides narrowed the difference between their perimeters. He also developed highly accurate calculations of the sine of one degree, a value useful in trigonometric tables. These were computational results, distinct from the accuracy of a particular observation.

Recording and checking results

Observation produced readings that had to be recorded, compared and converted into tabular form. Earlier astronomical works supplied methods and values to be tested. A discrepancy might arise from an instrument, a calculation or a copying error, so the mathematical and material parts of the work remained connected.

Manuscripts carried the results beyond Samarkand after Ulugh Beg’s death in 1449. The observatory’s surviving arc documents the scale of the original measuring apparatus; the tables and scholarly writings preserve the calculations that could continue to be studied elsewhere.

For a broader overview, see Timurid Culture and Arts.