论文标题
关于Tusi的立方方程分类及其与Cardano公式和Khayyam的几何解决方案的连接
On Tusi's Classification of Cubic Equations and its Connections to Cardano's Formula and Khayyam's Geometric Solution
论文作者
论文摘要
奥马尔·卡亚姆（Omar Khayyam）对立方方程的研究启发了12世纪波斯数学家Sharaf al-Din Tusi，研究了积极根源的数量。根据著名的数学历史学家Rased的说法，图西分析了五种不同类型的方程式的问题。实际上，所有立方方程都可以还原为{\ it tusi form} $ x^2-x^3 = c $。图西确定$（0,1）$上的最大$ x^2-x^3 $发生在$ \ frac {2} {3} $时，并在$ c = \ frac {4} {4} {27}Δ$，$δ\ in（0,1）$（$ c = \ frac {4} {27} {27} e（0,1）$（0,1）$（0,1）$（0,1,1） $（\ frac {2} {3}，1）$，忽略$（ - \ frac {1} {3}，0）$中的根。给定一个{\ IT还原表格} $ x^3+px+q = 0 $，当$ p <0 $时，我们表明它可以还原为Tusi form，$δ= \ frac {1} {1} {1} {2}+{3 \ sqrt {3}它遵循的是，只有$δ= - （\ frac {q^2} {4} {4}+\ frac {p^3} {27}）$是正面的。这给出了TUSI形式的$δ$与Cardano公式的$δ$之间的明确连接。因此，当$δ\ in（0,1）$中时，而不是以复数的数字使用Cardano的公式时，就可以迭代近似。另一方面，对于$ p> 0 $的简化表格，我们给出了Cardono公式的新颖证明。根据Hogendijk的说法，尽管Tusi对衍生物的使用最大的计算将最大的计算计算出来，但Tusi可能受Euclid的影响。在这里，我们显示图西形式的最大化器可以通过基本代数操作来计算。实际上，对于{\ it二维式形式}，$ x-x^2 =δ/4 $，图西的方法导致了二次公式的简单推导，与po-shen loh的教学方法相媲美。此外，我们得出了{\ it一般TUSI形式}的类似结果。最后，我们提出了Khayyam几何解的新型推导。结果补充了关于图西工作的先前发现，并揭示了有关解决立方方程的历史，数学和教学法的进一步事实。
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.
