have discovered his expressions for the maxima. The quantities which can be obtained from al-Tusi's conditions for the numbers of roots of cubic equations by subtracting one side of these conditions from the other is today called the discriminant of the cubic polynomials obtained by subtracting one side of the corresponding cubic equations from the other. Although al-Tusi always writes these conditions in the forms , , or , rather than the corresponding forms , , or , Roshdi Rashed nevertheless considers that his discovery of these conditions demonstrated an understanding of the importance of the discriminant for investigating the solutions of cubic equations. Sharaf al-Din analyzed the equation x3 + d = b⋅x2 in the form x2 ⋅ (b - x) = d, stating that the left hand side must at least equal the value of d for the equation to have a solution. He then determined the maximum value of this expression. A value less than d means no positive solution; a value equal to d corresponds to one solution, while a value greater than d corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development in Islamic mathematics, but his work was not pursued any further at that time, neither in the Muslim world nor in Europe. Sharaf al-Din al-Tusi's "Treatise on equations" has been described by Roshdi Rashed as inaugurating the beginning of algebraic geometry. This was criticized by Jeffrey Oaks who claims that Al-Tusi did not study curves by means of equations, but rather equations by means of curves (just as al-Khayyam had done before him) and that the study of curves by means of equations originated with Descartes in the seventeenth century. Astronomy Sharaf al-Din invented a linear astrolabe, sometimes called the "staff of Tusi". While it was easier to construct and was known in al-Andalus, it did not gain much popularity. Honours The main-belt asteroid 7058 Al-Ṭūsī, discovered by Henry E. Holt at Palomar Observatory in 1990, was named in his honor. Notes References Retrieved March 21, 2011 from Encyclopedia.com. External links (PDF version) 1130s births 1213 deaths 12th-century Iranian mathematicians 13th-century Iranian mathematicians Medieval Iranian astrologers 12th-century Iranian astronomers Astronomers of the medieval Islamic world 13th-century Iranian astronomers 12th-century astrologers 13th-century astrologers People from Tus, Iran 13th-century inventors 12th-century inventors