有关《九章算术》的一些讨论

JIUZHANG SUANSHU (NINE CHAPTERS ON THE MATHEMATICAL ART) AND LIU HUI'S COMMENTARY

文章分为５部分，第一部分指出“晋武库”并非“晋朝武库”而是“晋王武库”，从而对刘徽作注年代问题给出了一个关键性的新论点，极易肯定刘徽注完成于魏景元四年。第二部分针对今本《九章算术》中有争议的部分，指出S少广术、负笼术、宛田术、环田术、密率术、五家共井术等，本身无误，以前人们由于种种原因而对其作了不同的误解，亟待澄清。第三部分对体积理论作了详细讨论，提出《九章算术》时代应该已经有了成熟的棋验法并用以求得各立体体积的公式。第四部分讨论《九章算术》的方程，指出我国古算是在整数环内操作，中间结果不应该有有理数，认为方程术有毛病并提出校正意见；对方程章中麻麦问旧术提出新的校正意见，只对极少量的且极常见的错误作了改正。第五部分解释了为什么刘徽特别提出“麻麦与均输少广之章重衰积分皆为大事”。

This paper, which discusses various problems of Jiuzhang and Liu Hui's comments, is divided into five parts. The first part concerns Liu's biography. We interpret 'Jin' as a king and not a dynasty, so we have a new argument to affirm that Liu Hui completed his commentary in AD 263. The second part concerns various Shu of Jiuzhang. We show that: Wantian is a sector and not a part of sphere surface: Huantian needs not be a full ring (as what Liu Hui thought). Milushu is another formula to evaluate the area of huantian. Shaoguangshu is a complete method to get LCM without any flaw. The Fulongshu and the answer of Gongjing are quite correct. All of them were well established and were criticized heretofore only because of their being misunderstood. In the third part we discuss in detail the theory of volume in Jiuzhang and show that Jiuzhang had discussed the general yangmas and bienaos, never restricted in normal case. Besides, Jiuzhang had made use of 'qi' (chess), by means of which established the formulas about volumes. The fourth part concerns the theory of equations of Jiuzhang and we show that in particular, the 'fa' is essentially the determinant, hence it is one of the fundamental concepts. In Jiuzhang we should work in the integral domain, and so in it we frequently have to do something which could be avoided for rational fields. We correct anew one paragraph which was hitherto regarded as hard to amend. In the last part we consider what Liu Hui singled out as three major issues.