Magic squares in the tenth century. Two Arabic treatises by Anṭākī and Būzjānī (Q511298)

This book contains the English translations as well as transcripts of the original Arabic texts on magic squares by Alī ibn Muḥammad al-Anṭākī and by Abū al-Wafā' al-Būzjānī, who both lived in the 10th century. Al-Anṭākī's text is preceded by comments on the first book of number theory by Nicomachos. Some of these number theoretical propositions have simple counterexamples; proofs are not given except in a few cases, and when proofs are given they are often defective. In addition to constructions of magic squares, the text by al-Anṭākī also contains a section on determining ``hidden numbers'': someone is asked to pick two numbers \(x\) and \(y\) and perform some operations on it, such as computing \(\frac12[(x+y)^2-x^2+y^2]\), and reveal one of them. It is then explained that you get the hidden number upon dividing the result obtained by the revealed number. For a brief and informative survey on the history of magic squares in Islamic countries, see [\textit{A.~Dja'fari Naini}, Geschichte der Zahlentheorie im Orient: im Mittelalter und zu Beginn der Neuzeit unter besonderer Berücksichtigung persischer Mathematiker. Braunschweig: Klose; Hamburg: Univ. Hamburg, Fachbereich Mathematik (Diss.) (1982; Zbl 1380.01004)].