Ibn Kushna al-Qumī’s Treatise in Refutation of Passages from al-Karajī’s al-Kāfī fī al-ḥisāb

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Document Type : Research/Original/Reqular Article

Authors

1 Isfahan Mathematics House

2 Faculty Member, Institute of the History of Science, University of Tehran

10.22059/jihs.2025.401845.371838

Abstract

Muḥammad ibn Kushna al-Qumī, an Iranian mathematician active during the 4th–5th centuries AH (10th–11th centuries CE), composed a treatise entitled ʿUddat Masāʾil li-Ibn Kushna fī al-Radd ʿalā Mawāḍiʿ min Kitāb al-Kāfī li-l-Karajī. This work constitutes a critical examination of five sections from the geometric part of al-Karajī’s al-Kāfī fī al-ḥisāb. The topics addressed include: the extraction of an arc from its chord, the diameter of the circumcircle of regular polygons, the surface area of the sphere, the area of the inhabited quarter of the Earth, the surface area of the cone, the volume of the sphere, and methods for measuring ground elevation in the excavation of canals and qanāts—each supported by rigorous geometric reasoning. While most of Ibn Kushna’s objections are framed in purely geometric terms, in certain cases, such as the second problem, he challenges the general validity of propositions formulated by al-Karajī by offering counterexamples. In the third problem, he critiques an astronomical method of measurement, referring to a chapter in the third book of Kushyār’s Zīj jāmiʿ (on hay’a), where he identifies an error in the method of computation. This paper evaluates the soundness of Ibn Kushna’s criticisms of selected geometrical arguments in al-Karajī’s al-Kāfī fī al-ḥisāb from a mathematical perspective

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Journal for the History of Science

Articles in Press, Accepted Manuscript
Available Online from 20 October 2025

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