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.

ʿIlm-i ʿUqūd-i Abnīyih first time is named in ʾIrshād al-Qāṣid from Ibn al-Akfānī; but reflection on primary sources shows this science has older antecedent. There is no text in this science from Islamic period; but be said Ibn al-Haytham and Karajī had have compilations in this science. Primary sources in science classification in Islamic world help us to obtain the clearer concept from this science. Search in this texts shows ʿIlm-i ʿUqūd-i Abnīyih was main science connected to architecture in Islamic sciences. This research defines ʿIlm-i ʿUqūd-i Abnīyih was lower than theoretical geometry and upper than practical geometry.