麦克劳林关于方程实根最大绝对值下界的法则

Maclaurin’s Rule of the Lower Limit of the Maximum Absolute Value of Real Roots of Any Algebraic Equation

18世纪英国数学家麦克劳林对于判定代数方程实根的界限做出了重要贡献,这项工作记载于其著作《代数论》第二部分的第5章中。在该章的最后,麦克劳林直接断言了一条关于方程实根最大绝对值下界的法则,但是作者发现这条法则并不总是成立。按照古证复原的原则,作者对这条法则的构造过程进行了复原,从而澄清了麦克劳林原来的数学思想,指出他的错误在于将分母误认为方程的次数,但实际上它应该是根的两两之积的项数。最后,通过修正他的错误,论文给出了该法则的正确形式。

British mathematician Maclaurin in the 18th century made an important contribution to deter-mining the limits of real roots of algebraic equations. This work is recorded in the fifth chapter of the second part of his book A Treatise of Algebra. At the end of this chapter, Maclaurin directly as-serts a rule about the lower limit of the maximum absolute value of real roots of any algebraic equation, but we find that this rule is not always true. According to the principles of recover para-digm, we have restored the deductive process of this rule, thus clarified Maclaurin’s original mathematical thought and pointed out that his mistake lies in mistaking the denominator for the degree of the equation, but in fact it should be the number of terms of the product of any two roots. Finally, we have amended his mistake and given the correct form of this rule.