卡尔达诺如何利用“和的法则”列方程

How Cardano Use “the Rule for a Sum” to Establish Equations

摘要: 卡尔达诺的《大术》是讨论方程求解的历史名著。在这本书的第35章，他在求解具体问题时利用“和的法则”来列方程。其中，对于10个问题所导致的方程，他并没有给出详细的推导步骤。我们发现，按照现代方法列出的方程与卡尔达诺的方程往往并不相同。那么，他是如何利用“和的法则”列出这些方程的？通过对卡尔达诺列方程的思路进行分析，并利用他已经知道但未予明示的平方和法则和平方差法则，我们复原了这10个方程的推导过程，从而比较圆满地解决了上述问题。

Abstract: Cardano’s “Ars Magna” was a historical masterpiece in discussing the solution of equations. In Chapter 35 of this book, he used “the rule for a sum” to establish equations when solving concrete problems. Among these problems, there were 10 equations which he gave out directly without any detailed steps for deriving them. We find that the equations established according to modern methods are often different from those of Cardano’s. So, how Cardano used the “the rule for a sum” to establish these equations? By analyzing Cardano’s own idea of establishing equations, and using the rules for sums or difference of squares that he had already known but had not explicitly stated, we have restored the derivation process of these 10 equations and solved the above problem successfully.