Extracting Square Roots Made Easy: A Little Known Medieval Method

In 1901 the Swiss historian of mathematics and Arabist Heinrich Suter (1848-1922) published in the journal Bibliotheca mathematica the German translation of an Arabic mathematical treatise composed by (as the author himself wrote) “Master Abu Zakariya Muhammad ibn Abd Allah known under the name of el-Hassar,” in which we find the earliest known description of an iterative process for extracting square roots yielding closer and closer approximations to the real value. Although this ingenious yet basically simple way to calculate square roots – completely different from our method today – was taught later by the outstanding Italian mathematicians Luca Pacioli (ca. 1445-1517), Hieronimo Cardano (1501-1576), Nicolo Tartaglia (1499/1500-1557), and Pietro Antonio Cataldi (1548-1626), it is largely unknown today. But it is worthy of study, and we present it in this article.