程之骥《开方用表简术》补记

Some Notes on CHENG Zhiji's the Simple Method of Extraction with Tables(KaiFang YongBiao JianShu)

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·少广》章.后经宋元发展为"立成释锁"和"增乘开方"算法,这是一种解决一元高次方程的一个正实数根的有效方法,对于开方不尽而有奇零的情况,如根为无理数则"命分"得近似值,或"续开"直到满足要求精度;分数(小数)根则可"续开"得近似或准确值,"之分法"则可得准确值.明朝"增乘开方"算法失传,珠算"立成释锁"开方法一枝独秀.清后期李锐、汪莱和焦循等,重新发现并使用"增乘开方"算法.华蘅芳在其《开方古义》中误以为发现了古代"增乘开方"算法,实际上是"增乘立成"开方法,程之骥则简而化之.因程之骥书中有缺漏部分,今以算补校.

Extraction method is one of the most advanced and mature content development in Chinese traditional mathematical algorithm,first seen in" nine chapters".And it were developed into two methods of" Zengchengkaifang" and" Lichengshisuo",a simple method to solve the equations with one unknown only for one positive solution,sometimes approximate root,after the Song and Yuan dynasties.In Ming dynasty,the methods ofZengchengkaifang" was lost,actually it's better than the other one.Fortunately,it is lost and found in Qing dynasty,and some scholars devoted to study it and got some good results,in which CHENG Zhiji created the simple method of extraction with tables based on the works of Hua Hengfang.Some note were made for his book,and some assessments are put tentatively.