【作者】 宋华；

【导师】 郭世荣；

【作者基本信息】 内蒙古师范大学 ， 科学技术史， 2003， 硕士

【摘要】 19世纪后期是中国数学由传统向近现代转化的一个重要时期，微积分是近代数学乃至大部分自然科学的最基础的理论，这个理论是从1859年出版的译著《代微积拾级》中介绍到中国来的。但是，直到清末，真正对西方传入中国的以微积分为主的近代数学知识有深刻体会的中国数学家还并不多。微积分、解析几何、代数学等知识在中国落地扎根并不是一帆风顺的，而是经历了一个相当长的时期。对晚清数学家来说，面临着新的机遇，同时存在着巨大的挑战和困惑。 19世纪中算家对《代微积拾级》、《微积溯源》、《决疑数学》等西方译书的学习、理解以及应用，是中国数学由传统向近代转化过渡的一个十分重要的方面。研究晚清中算家对微积分的学习情况、理解程度以及应用能力，无疑对于研究中国数学的近代化历程具有十分重要的意义。这是中国数学史研究的一个重要课题。目前，在这个课题上的研究工作还较少，只有为数不多的一些零散论述。 作为内蒙古自治区教育厅重点项目“中国数学近代化之历程研究”的一个组成部分，在本文中，我们选定清末数学家夏鸾翔的《万象一原》作为研究对象，主要目的是通过对《万象一原》的研究来认识晚清数学家是如何学习和使用微积分的，并探讨他们对微积分的掌握情况和理解程度。 《万象一原》(1862)是夏鸾翔在学习了《代微积拾级》(1859)之后完成的一部著作，这是夏鸾翔的最后一部数学著作，同时也是我国数学家自己写的第一部应用微积分知识的著作。书中利用微积分这个有力的工具解决戴煦、项名达等人在求弧长、面积和体积时所遇到的问题和困难，有一些新的进展。以往对微积分在中国的传播史和夏鸾翔的研究，虽然也有一些工作把夏鸾翔的研究和微积分联系在了一起，但是还存在着不少问题和不清楚的地方。一方面，前人涉及到的仅是《万象一原》中的一小部分例题，所得结果不能完全反映夏氏对微积分的认识程度和使用情况，另一方面没有清晰地把《万象一原》和《代微积拾级》联系起来。 本文由五部分组成，各部分的内容大致如下： 一、阐述本选题的意义，介绍前人在这个课题上的已有成果，说明本文的主要研究内容和思想。 夏莺翔对微积分的学习与使用—《万象一原》内容分析 二、介绍与本文有关的背景材料，包括:微积分传入中国的过程;夏莺翔的生平履历;他的前辈数学家们在他所研究的问题上的相关研究成果;《万象一原》的内容和结构;关于清末研究《万象一原》的两本数学著作。 三、用现代形式整理了《万象一原》中的130多个公式，说明了这些公式的几何意义(四个除外)，校验了各公式的正误情况，并与《代微积拾级》进行了对比。 四、方法分析部分，通过对书中部分典型例题的详细讨论，分析和说明夏氏是如何得到这些公式的。 五、总结全文，分析夏氏对微积分的理解认识程度，说明《万象一原》的方法、贡献和意义。 本文的新工作主要有:说明了《代微积拾级》与另二种译著《微积学教科书》(1904)和《微积学》(1912)的底本之间的关系;首次研究了《万象一原校勘记》和《万象一原演式》二书;整理了《万象一原》的全部公式(前人曾分析过其中的极少部分)，并分析了这些公式的推导过程;从中算近代化的角度分析了夏氏的微积分水平。更多还原

【Abstract】 With the publication of the Dai Wei Ji Shi Ji (DWJ), a translation of Loomis’ textbook Elements of Analytical Geometry and of the Differential and Integral Calculus (1851), in 1859, the differential and integral calculus was introduced into China. But until the end of the Qing dynasty, only a few Chinese mathematicians had mastered the theory of differential and integral calculus. When they studied and applied the theory of calculus, they confronted both great challenge and opportunity. Therefore, it took a long period for calculus, analytical geometry and algebra to take root in China.Chinese mathematics began its transition from traditional mathematics to modern mathematics in the 19th century. It is an important step of the transition for Chinese mathematicians to study and to use the translations of western mathematical textbooks on calculus, analytical geometry, algebra, and probability, such as DWJ, Wei Ji Su Yuan, Jue Yi Shu Xue. Therefore, clarifying how Chinese mathematicians studied, understood, and applied the calculus in the late Qing is necessary for the study of the transition. It is an interest topic in the study of the history of mathematics in China.The present paper, as a component of a research project: "Studies of the Transition of Chinese Mathematics from the Traditional to Modern One" supported by Inner Mongolia Education Bureau, deals with mathematical treatise Wan Xiang Yi Yuan (WXYY) written by Xia Luanxiang (1823-1864) in 1862. The treatise is the last mathematical work of Xia, as well the first treatise applying the differential and integral calculus after appearance of the DWJ in China. Xia Luanxiang was a celebrate mathematician in the late Qing. In this thesis the author gives a comprehensive study of WXYY, attempting to get some idea about how Xia mastered and applied the calculus.The paper consists of five parts. The content of each part is as follows:Part 1 Explaining the meaning of the dealt topic and the research contents and purpose of the thesis, and surveying the related research results.Part 2 Introducing the some background materials, including the spreading process of calculus in China, Xia’s bibliography, the topic of the WXYY and relevant researches done by other mathematicians before him, structure of WXYY, and two treatises studying WXYY which appeared in 1901 and 1902 respectively.Part 3 Rewriting and classifying all 136 formulas, which were written in traditional terminology, in today’s terminology; explaining the geometrical meanings of the formulas; checking all the formulas and correcting all kinds of mistakes appearing in the original text; comparing some of them with those of DWJ,Part 4 Analyzing some of the typical formulas and explaining how Xia obtained them.Part 5 Summing up the results of the paper; evaluating the degree of Xia’s mastery of the calculus, his method, and the achievement in the WXYY.The paper adds some new results to the study of the history of mathematics in China. Firstly, the relation between the DWJ and other two translations of Loomis’s text towards the end of the Qing is discussed. Secondly, two treatises written in 1901 and 1902 respectively are studied for the first time. Thirdly, all formulas appeared in WXYY are classified and rewritten in today’s terminology (only several of them were analyzed before), and the deriving methods of them are analyzed. Lastly, Xia’s level of using calculus is evaluated from the perspective of transition of traditional Chinese mathematics to the modern one.更多还原