On Tusi's Classification of Cubic Equations and its Connections to Cardano's Formula and Khayyam's Geometric Solution

Abstract

Omar Khayyam's studies on cubic equations inspired the 12th century Persian mathematician Sharaf al-Din Tusi to investigate the number of positive roots. According to the noted mathematical historian Rashed, Tusi analyzed the problem for five different types of equations. In fact all cubic equations are reducible to a form {\it Tusi form} $x^2-x^3=c$. Tusi determined that the maximum of $x^2-x^3$ on $(0,1)$ occurs at $\frac{2}{3}$ and concluded when $c=\frac{4}{27} \delta$, $\delta \in (0,1)$, there are roots in $(0, \frac{2}{3})$ and $(\frac{2}{3},1)$, ignoring the root in $(-\frac{1}{3},0)$. Given a {\it reduced form} $x^3+px+q=0$, when $p <0$, we show it is reducible to a Tusi form with $\delta = \frac{1}{2} + {3\sqrt{3} q}/{4\sqrt{-p^3}}$. It follows there are three real roots if and only if $\Delta =-(\frac{q^2}{4}+\frac{p^3}{27})$ is positive. This gives an explicit connection between $\delta$ in Tusi form and $\Delta$ in Cardano's formula. Thus when $\delta \in (0,1)$, rather than using Cardano's formula in complex numbers one can approximate the roots iteratively. On the other hand, for a reduced form with $p >0$ we give a novel proof of Cardono's formula. While Rashed attributes Tusi's computation of the maximum to the use of derivatives, according to Hogendijk, Tusi was probably influenced by Euclid. Here we show the maximizer in Tusi form is computable via elementary algebraic manipulations. Indeed for a {\it quadratic Tusi form}, $x-x^2=\delta/4$, Tusi's approach results in a simple derivation of the quadratic formula, comparable with the pedagogical approach of Po-Shen Loh. Moreover, we derive analogous results for the {\it general Tusi form}. Finally, we present a novel derivation of Khayyam's geometric solution. The results complement previous findings on Tusi's work and reveal further facts on history, mathematics and pedagogy in solving cubic equations.