| 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.
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