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Mathematical Treasure: Commentary on the Fundamental Theorems

Frank J. Swetz (The Pennsylvania State University)

Sharḥ ashkāl al-taʾsīs (Commentary on the Fundamental Theorems, or Commentary on the Forms of Foundation) was written in 1413 [Youschkevitch, p. 123] by the Islamic scholar Qāḍī Zāda Mūsā ibn Muḥammad, also known as al-Rūmī ("the Roman" or "the Byzantine") because he was born in Asia Minor. His year of birth is not known; he died in 1436. The University of Michigan Library Catalog describes the manuscript (Islamic Manuscript 727) more specifically as a “commentary on the Ashkāl al-taʾsīs [The Fundamental Theorems], a compendium of geometry according to Euclid by Muḥammad ibn Ashraf al-Samarḳandī.” Al-Samarḳandī lived during the 13th century. The copy shown here was written sometime after Qāḍī Zāda wrote his commentary (in about 1413) and before 1612.

A discussion of the area of the regions resulting from the bisection of a rectangle by its diagonal (Euclid I.43):

A rendering of the “Wind Mill Proof” of the Pythagorean Theorem, one popular with Islamic authors:

The images above are supplied through the courtesy of the University of Michigan Libraries. You may use them in your classroom; for permission to use them for any other purpose, please contact the University of Michigan Libraries. The full manuscript is available via the HathiTrust Digital Library as

Record 002583102, Sharḥ ashkāl al-taʾsīs : manuscript, [between 1400? and 1611], HathiTrust Digital Library.

Besides the two library catalog webpages (University of Michigan and HathiTrust) cited above, information for this article was obtained from the following source:

Adolf P. Youschkevitch, Les Mathématiques Arabes (VIIIe – XVe siècles), Paris: Vrin, 1976.

See pages from two other copies of Qāḍī Zāda’s commentary on al-Samarḳandī’s The Fundamental Theorems (based on Euclid’s Elements) in Mathematical Treasures - Qāḍī Zāda’s Geometry and Mathematical Treasure: Qadi Zada’s Commentary on The Forms of Foundation.

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Commentary on the Fundamental Theorems," Convergence (May 2017)