Mathematical works. Algebra and geometry in the XIIth century. Vol. I, II. Dual French and Arabic text. Text established and translated by Roshdi Rashed (Q2736397)

These two volumes contain Arabic texts and French translations of the extant mathematical works of the 12th-century Iranian mathematician Sharaf al-Din al-Tusi. He is not the same as the famous Iranian mathematician and astronomer Nasir al-Din al-Tusi, who lived in the 13th century. The works by Sharaf al-Din include a brief text on the asymptotes of the hyperbola, a short treatise on an elementary problem in the division of areas, and a very long text, which is usually called the Algebra (the original title is lost). In his Algebra, Sharaf al-Din improves the treatment of cubic equations in the Algebra of his Iranian predecessor `Umar al-Khayyami (11th century). The medieval Islamic mathematicians were unable to solve cubic equations algebraically. Because they did not recognize zero and negative coefficients, and only worked with positive roots, they distinguished 17 different types of cubic equations. For all these types, `Umar al-Khayyami showed how a root can be constructed geometrically by means of intersecting conic sections. Sharaf al-Din divides the cubic equations into two groups.NEWLINENEWLINENEWLINEThe first group (discussed in Volume 1 of the work under review) consists of the equations which for all (positive) choices of the coefficients have a positive root \(x\), such as \(x^3+ax^2=c,\) where \(a,c > 0.\) For these equations, Sharaf al-Din gives the same construction by means of conic sections as al-Khayyami, but Sharaf al-Din adds a proof that a point of intersection exists and an algorithm for the numerical approximation of \(x\). This algorithm is a variant of the so-called method of Ruffini and Horner, which seems to be of ancient Chinese origin. Sharaf al-Din does not mention that the equation \(x^3+bx=ax^2+c\) can have two or three positive roots for suitable choices of the coefficents.NEWLINENEWLINENEWLINEThe second group (discussed in Volume 2) consists of the equations that do not always have a positive root, such as \(x^3+c=bx\). In these cases, al-Khayyam only remarked that the root \(x\) exists if the conic sections in his construction intersect. Sharaf al-Din discusses the necessary and sufficient conditions, in terms of \(a,b\) and \(c\), for the existence of a positive root \(x\) as well as the number of positive roots \(x\). His results are as follows in modern notation. Write the equations as \(f(x)=c\). Sharaf al-Din finds a magnitude \(m\) such that the equation has zero, one or two solutions if \(c\) is greater than, equal to or less than \(f(m)\), respectively. In modern terms \(f'(m)=0\), but Sharaf al-Din does not use the modern derivative. For \(c<f(m)\), the equation has one root \(x_{1}>m \) and another root \(x_{2}<m\). Sharaf al-Din constructs these roots by substituting \(y=x_{1}-m\) and \(z=m-x_{2}\); he shows that \(y\) and \(z\) satisfy the equations \(y^3+py^2=d\) (1) and \(z^3+d=pz^2\) (2), for \(d=f(m)-c\) and for a certain \(p>0\) which he defines in terms of the coefficients. The root \(y\) of (1) had been constructed and approximated in the first part of the Algebra, so \(x_{1}\) can also be constructed and approximated. Sharaf al-Din proves the existence of \(z\) by showing that \(z-y\) satisfies a quadratic equation with coefficients dependent on \(p, d\) and \(y\). He then gives an algorithm for the approximation of \(z\) from (2). In the introduction to this book, Rashed argues that several concepts from 17th century and later mathematics, such as the derivative, affine transformations, and cubic curves, are implicit in the work of Sharaf al-Din. The reviewer has proposed an alternative interpretation of Sharaf al-Din's Algebra in the framework of traditional Islamic geometry in [\textit{J. P. Hogendijk}, Hist. Math. 16, No. 1, 69-85 (1989; Zbl 0672.01005)], with a summary of the Algebra in English. Rashed's views have been defended by \textit{N. Fares} [Hist. Sci. (2) 5, No. 1, 39-55 (1995; Zbl 0838.01007); Arab. Sci. Philos. 5, No. 2, 140, 142, 219-237 (1995; Zbl 0846.01002)]. For a comparison between different interpretations see [\textit{J. L. Berggren}, J. Am. Orient. Soc. 110, No. 2, 304-309 (1990)].