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Friedrich Katscher (Vienna University of Technology)



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.




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.


Later, al-Hassar's method was taken up again in the hand-written Treatise of Arithmetic by Aboûl Hasan Ali ben Mohammed Alkalçadi, 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.


