论土司对三次方程的分类及其与卡尔达诺公式和海亚姆几何解的联系
On Tusi's Classification of Cubic Equations and its Connections to Cardano's Formula and Khayyam's Geometric Solution
摘要：
Omar Khayyam对三次方程的研究启发了12世纪波斯数学家Sharaf al-Din Tusi研究正根的数量。根据著名数学历史学家拉希德的说法，土司分析了五种不同类型的方程的问题。事实上，所有的三次方程都可以简化为一种形式{ \it Tusi form} $x^2-x^3=c$。Tusi确定$(0,1)$上$x^2-x^3$的最大值发生在$\frac{2}{3}$，并得出结论，当$c=\frac{4}{27} δ$、$δ\in (0,1)$、有根在$(0, \frac{2}{3})$和$(\frac{2}{3},1)$时，忽略根在$(-\frac{1}{3},0)$。给定一个{ \it约化形式} $x^3+px+q=0$，当$p <0$时，我们证明了它可约化为具有$δ= \frac{1}{2} + {3\sqrt{3} q}/{4\sqrt{-p^3}}$的土司形式.有三个实根当且仅当$Δ=-(\frac{q^2}{4}+\frac{p^3}{27})$为正。这给出了土司形式的$δ$和卡尔达诺公式中的$Δ$之间的明确联系。因此，当$δ\in (0,1)$，而不是使用卡尔达诺的公式在复数可以近似的根迭代。另一方面，对于一个简化的形式与$p >0$，我们给出了一个新的证明Cardono的公式。虽然拉希德属性土司的计算最大的使用衍生物，根据霍根迪克，土司可能是受欧几里德。在这里，我们表明，在土司形式的最大化是可计算的，通过初等代数操作。事实上，对于{ \it二次土司形式}，$x-x^2=δ/4$，土司的方法结果在一个简单的推导二次公式，可比的教学方法的宝沉洛。此外，我们得到类似的结果{ \it一般土司形式}。最后，我们提出了一个新的推导海亚姆的几何解决方案。结果补充了以前的调查结果土司的工作，并揭示了进一步的事实，历史，数学和教育学在解决三次方程。
英文摘要：
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 $c=\frac{4}{27} δ$, $δ\in (0,1)$, 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 $δ= \frac{1}{2} + {3\sqrt{3} q}/{4\sqrt{-p^3}}$. It follows there are three real roots if and only if $Δ=-(\frac{q^2}{4}+\frac{p^3}{27})$ is positive. This gives an explicit connection between $δ$ in Tusi form and $Δ$ in Cardano's formula. Thus when $δ\in (0,1)$, 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}, $x-x^2=δ/4$, 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.
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