Miftāḥ al-ḥisāb

work by al-Kāshī

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July 16, 2026

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The Miftāḥ al-ḥisāb (Key of Arithmetic) is a significant work by the Persian mathematician and astronomer Jamshīd al-Kāshī, completed in 1427. This encyclopedic text provides instructions for solving a wide array of problems in arithmetic, surveying, and astronomy. Al-Kāshī defined arithmetic as the science of finding numerical unknowns from known quantities. The Miftāḥ al-ḥisāb is renowned for its pedagogical clarity, leading to numerous copies being made over centuries. Al-Kāshī was a prominent scholar in the Islamic world, working in Samarkand under the patronage of Ulugh Beg, a grandson of Timur. His work contributed to the advancement of mathematics during the Islamic Golden Age.1

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al-Kāshī
Life in Samarkand
Al-Kāshī’s best-known work is the Miftāḥ al-ḥisāb (“Key of Arithmetic”), completed in 1427 and also dedicated to Ulūgh Beg. This encyclopedic work instructs in the solution of a wide range of problems from astronomy, surveying, and finance through the use of arithmetic—defined by al-Kāshī as “the science consisting of basic…
Read More
Al-Kāshī’s best-known work is the Miftāḥ al-ḥisāb (“Key of Arithmetic”), completed in 1427 and also dedicated to Ulūgh Beg. This encyclopedic work instructs in the solution of a wide range of problems from astronomy, surveying, and finance through the use of arithmetic—defined by al-Kāshī as “the science consisting of basic…
Read More