Extracting Square Roots Made Easy: A Little Known Medieval Method - Historical Background

Friedrich Katscher (Vienna University of Technology)

All we know about Abu Zakariya (in Hebrew manuscripts, Abu Bakr) Muhammad ibn Abu Abd Allah Ayyash el- or al-Hassar (the rush mat maker) is that he lived in the 12th century in the Islamic West, in the Maghreb (today's Morocco, Algeria and Tunisia) or in Andalus (southern Spain). As often happens, no original manuscript by the author has survived, but five later copies do survive:

The chapter on the square roots is not contained in all five. Some copies bear the title Book of Proof and Recall. The copy in Gotha has no title. It has 128 leaves.

In the chapter on fractions we read: “but if you have to represent a fraction, then write the denominator under a [horizontal]* line and above each its mentioned part.” This is the first known use of the fraction bar. A little later Leonardo Pisano (“Fibonacci”) became the first European mathematician to employ it in his Liber ab(b)aci of 1202.

Al-Hassar's book is supposed to be the oldest mathematical text of the Maghreb and Moslem Spain. He wrote a second larger work with the title, The large book or The complete book on the art of number, of which the first part, Book I, was found in 1986, the first pages missing, in the Library Ibn Yusuf in the Moroccan city of Marrakesh. Its contents were described in 1987 by Mohamed Aballagh and Ahmed Djebbar. The second part, Book II, probably is lost.

Al-Hassar does not claim to be the originator of the new arithmetic propositions he expounds. On the contrary, he makes it clear at the outset that the opposite is true:

And everything that I have put together, described and explained in this book, stems from the statements of older scholars, and I have drawn it from the books of the ancestors, I have collected it, commented on it, and found and derived it thanks to their sound reasonings.

Unfortunately most of the “books of the ancestors” (or, rather, manuscripts of the ancestors) have been lost.

Only a few decades later, in his 1202 Liber Ab(b)aci, Leonardo Pisano used al-Hassar’s method to calculate the first and second approximations of the square root of 10 and of larger numbers. Pisano did not mention al-Hassar’s name, and probably had learned the method from someone else.

In 1271 the Jewish physician, author, and outstanding translator Moses ben Samuel ibn Tibbon, who lived in Montpellier in southern France, translated al-Hassar's text into Hebrew under the title Sefer ha-heshbon (Book of arithmetic), of which there is one manuscript in the Vatican and one in Oxford.

Later, al-Hassar's method was taken up again in the hand-written Treatise of Arithmetic by Abo├╗l Hasan Ali ben Mohammed Alkal├žadi, who was born in Spain in 1412 and died in Tunisia in 1486 (Woepcke, 1859).

The first printed book that contained all the mathematical knowledge of its time was the Summa de Arithmetica Geometria Proportioni et Proportionalita of Luca Pacioli of 1494 (reprinted in 1523), and of course you find al-Hassar's procedure in it (Woepcke, 1874). From 1556 to 1560 Nicolo Tartaglia published his mathematical encyclopedia, the General Trattato di Numeri et Misure (General Treatise of Numbers and Measurements), and he wrote in the second volume in the chapter “Regola di saper sempre approssimarsi piu nelle radici sorde” (“Rule for knowing how to approach always closer the irrational roots”) that “those ancient Arabs, most expert investigators of these practices, ... sought yet another rule for an increasingly close approximation of the true value, & ad infinitum.” This means Tartaglia knew that the rule did not originate in Europe.

Al-Hassar’s method was treated also in 1539 in the Practica arithmetice of Hieronimo Cardano, in 1559 in the Logistica (Arithmetic) of the Frenchman Ioannes (Jean) Buteo, and in 1613 in the Trattato del modo brevissimo di trouare la radice quadra delli numeri (Treatise on the shortest method to find the square root of numbers) (Woepcke, 1874), in which the author Pietro Antonio Cataldi (1548-1626) published his discovery of continued fractions.

Beginning in the 16th century, arithmetic books in most countries adopted the present-day method for the extraction of square roots, the one that is still currently taught in schools. When more accurate results were needed, pairs of zeros (\(00\) or \(0000,\) etc.) were added to the radicand. This is still done today. Things were different in Italy, however. Italian arithmetic books applied the present-day method only to square numbers. For non-square numbers, the method described by the great Italian mathematicians Pacioli, Cardano, and Tartaglia was retained in textbooks and reprints until the very end of the 18th century. Thereafter, the method we trace back to al-Hassar fell into oblivion even in Italy.


* Throughout this article, material [in brackets] has been inserted by the author, in order to assist the reader.