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Ghiyāth al-Dīn Jamshīd al-Kāshī

Persian Mathematician and Astronomer of the Timurid Renaissance

Ghiyāth al-Dīn Jamshīd al-Kāshī (c. 1380 – 22 June 1429 CE) was a Persian mathematician and astronomer born in Kashān, Persia, and later based at the Samarkand Observatory under Ulugh Beg. He is celebrated for his extraordinary numerical precision in calculating π and trigonometric values, and for his influential mathematical writings such as the Miftāḥ al-ḥisāb.:contentReference{index=0}

Key Mathematical and Astronomical Contributions

• Treatise on the Circumference (Risālah al-muḥīṭīyya, 1424): Computed \(2\pi\) to 9 sexagesimal places—equivalent to 16 decimal places of accuracy.:contentReference{index=1}
• The Key of Arithmetic (Miftāḥ al-ḥisāb, 1427): A five-book encyclopedic work covering integer and fraction arithmetic, astronomy, geometry, and algebra.:contentReference{index=2}
• Treatise on the Chord and Sine (Risālah al-watar waʾl-jaib): Calculated \(\sin 1^\circ\) to exceptional accuracy; used advanced iterative methods akin to Newton’s method.:contentReference{index=3}
• Astronomical Tables (Khaqani Zij): Included highly accurate sine tables (to four sexagesimal digits per degree) and coordinate transformations.:contentReference{index=4}
• Planetary Equatorium: Invented analog computing instruments—such as the Plate of Conjunctions and Plate of Zones—for predicting planetary positions and eclipses. His equatorium was unique in also determining planetary latitudes.:contentReference{index=5}

Innovations and Significance

Al-Kāshī delivered the most accurate approximations of π and trigonometric values of his era. His computational methods, especially for roots and sine calculations, preceded similar discoveries in Europe by centuries. His instruments represented a pinnacle in pre-telescope analog computation.:contentReference{index=6}

Legacy and Influence

• Known as “the second Ptolemy,” he played a central role in the Samarkand scientific renaissance under Ulugh Beg.:contentReference{index=7}
• Miftāḥ al-ḥisāb became a foundational mathematical text for scholars across many disciplines for centuries.:contentReference{index=8}
• Although his work remained obscure in Europe for a long time, modern historians recognize his pioneering contributions to numerical analysis, trigonometry, and astronomical instrumentation.:contentReference{index=9}