开方术与多个正根方程:从“可知”到“不可知”

The"Kaifangshu"(Extract of a Root)and the Equations with the Several Positive Roots:from"Kezhi"(Equations with One Positive Root)to"Bukezhi"(Equations with Multi-Positive Roots)

开方术是中国传统数学中发展较为完善和成熟的内容之一,最早见于《九章算术》和《少广》章.后经宋元发展为“立成释锁”和“增乘开方”算法,是一种解决一元高次方程的一个正实数根的有效方法.不知是巧合还是人为抑或是算法本身的局限,中国古代的高次方程几乎都只有一个正根;而两个正根的方程作为不和谐的声音表现为赵爽二次方程求解的隐喻,刘益、杨辉的视而不见和秦九韶、李冶的不选之择.直到汪莱指出对秦九韶和李冶“不可知为可知”的错误,多正根的方程即“不可知”才进入算家的视野.李锐以前对于有多个正根的方程没有有效的解决方法.为此,李锐给出新的步法,用“代开法”求出多个正根.而就多个正根的方程而言,用基于新步法的“正负开方术”是求解方便且计算效率高的方法.

Extraction method is one of the most advanced and mature content in Chinese traditional mathematical algorithm,first seen in"Nine Chapters".And it was developed into two methods of"Zengcheng kaifang"and"Lichengshisuo",an effective method to solve the equations with one unknown only for one positive solution,sometimes approximate root,after the Song and Yuan dynasties.However,the equations with two positive solutions did appear in the Chinese history of mathematics which showed as Zhaoshuang’s mtaphor of,Liuyi and Yanghui’s ignoring and Qinjiushao and Liye’s unreasoning.And when Wanglai pointed that Qinjiushao and Liye’s mistake taking the"Bukezhi"as"Kezhi",the equation with the several positive roots were embraced by the Chinese mathematicians.Some scholars devoted to study it and got some good results in which we find the method called"Bufa",which first used by LiRui helping to solve the equation with the several positive roots.By"Bufa",LiRui can evaluate that how many and approximate values especially the first digital number of the equation,then he’ll use"Daikai"method to solve each positive root.As for the equations with multiple positive roots,"Zhengfu kaifang"method based on the new"Bufa"is a convenient and efficient method.