秦九韶“大衍总数术”中问数化定数算法解析

秦九韶在“大衍总数术”中化问数为定数有两条途径:一是将诸问数直接两两连环求等；一是将诸问数先求总等,存一(位)约众(位),然后再连环求等.连环求等时的根本原则是“约奇弗约偶”,其中的“奇”、“偶”系指等数的个数的单、双.“约奇弗约偶”就是在化约时约含有奇数个等数的问数而不约含有偶数个等数的问数,目的是使约后的两数互素.如果化约后得到的两数不互素,即属有续等的情形,此时要用“以续等约彼则必复乘此”的方法再次化约.若约后仍有续等,则要继续用此方法化约,直至求得定数.诸问数的排列顺序以及求等化约的先后次序不会影响计算结果的正确性.两种求定数的途径异曲同工,秦氏的算法是具有一般性的通法.

Analysis of the Algorithm of Reducing Wenshus to Dingshus in Qin Jiushao's Dayan Zongshu Method

In his Dayan Zongshu Method,Qin Jiushao shows us two channels of reducing wenshus to dingshus.One is to join all the wenshus by twos and then find one by one the dengshus(greatest common divisors,GCDs) of those pairs.The other way is to find the zongdeng(GCD of all the wenshus) first,select one wenshu and keep it and make other wenshus divide the zongdeng to get quotients,then set up the selected wenshu and the quotients,join them by twos and find the dengshus of the different pairs.The fundamental principl...