From ancient times to the present, mathematics, as a form of knowledge, has undergone a variety of development and change processes. Before the end of the Qing Dynasty, there were two climaxes in the introduction of Western mathematics into China. In 1582 (the tenth year of the Ming Dynasty), the Italian missionary Matteo Ricci (1522-1610) came to China, and he collaborated with Xu Guangqi to translate the "Geometric Origins", which introduced Euclid geometry to China, which is considered by scholars to be "the beginning of the Westernization of Chinese mathematics".
However, Matteo Ricci and Xu Guangqi only completed the first 6 volumes of the Geometric Primitives, and the translation of the last 9 volumes of the Geometric Primitives was delayed for more than 200 years before being completed by the protagonist to be introduced today, the Qing Dynasty mathematician Li Shanlan, in collaboration with the British sinologist Alexander Wylie (1815-1887). They played an important role in the second introduction of Western mathematics into China in the late Qing Dynasty. Since then, traditional Chinese mathematics has gradually been westernized and integrated with international mathematics.
Promoter of the Westernization of traditional Chinese arithmetic
Li Shanlan (1811-1882), a famous mathematician, educator and translator in the late Qing Dynasty, was originally named Xinlan, Zi Ren Shu, Qiu Xiu, a native of Haining, Zhejiang. He liked mathematics since he was a child, and when he was 10 years old, he "read the ancient "Nine Chapters" on the shelf, stole it, thought that he could do it without learning, and from then on he calculated well." At the age of about 15, he taught himself the first six volumes of the "Geometric Origins", "Measuring the Round Sea Mirror", "Pythagorean Cutting Circle", "Four Yuan Jade Jian" and other mathematical works. He is also a long and active writer.
Figure 1 Li Shanlan (1811-1882)
Euclidean geometry's rigorous logical system and clear mathematical reasoning are very different from the traditional Chinese mathematical ideas that emphasize practical solutions and computational techniques, and have their own characteristics and strengths. On the basis of the Nine Chapters of Arithmetic, Li Shanlan also absorbed the Western mathematical knowledge system in the "Primitive Geometry", which made his mathematical attainments more and more profound.
While studying traditional Chinese mathematical theories, Li Shanlan also actively translated Western scientific works. In 1857, Li Shanlan and the British sinologist Alexander Wylie (1815-1887) jointly completed the translation of the last 9 volumes of Euclid's Geometric Primitives. At the same time, he also co-translated with Villiers the 18 volumes of "Tantian", which introduces modern Western astronomical theory, 18 volumes of "Algebraics" that introduces Western calculus theory, and 13 volumes of "Algebra", which introduces Western symbolic algebra theory. In addition, Li Shanlan co-translated with Joseph Edkins (1823-1905) the 20 volumes of "Heavy Studies" that introduced Western mechanical theory, and with Alexander Williamson (1829-1890) translated 8 volumes of "Botany" introducing Western botany theory.
It is worth mentioning that Li Shanlan also co-translated Newton's classic "Mathematical Principles of Natural Philosophy" (translated as "Neptune Mathematics") together with Wei Lieali and Fu Laner (1839-1928), but unfortunately it was not translated and could not be published.
In translating Western writings, Li Shanlan coined many important Chinese mathematical terms. "Algebra", "function", "equation", "differentiation", "integral", "series", "plant", "cell", etc., were all used by him and introduced into the Chinese academic system. The scientific terms he chose Chinese were not only relevant and easy to understand, but also elegant. These terms were not only circulated in China, but also spread east to Japan and are still used today.
Li Shanlan was an important figure in the process of transforming traditional Chinese arithmetic into modern mathematics, and he contributed to both traditional Chinese arithmetic and the introduction of Western mathematical knowledge. His mathematical work "Zegu Xi Zhai Arithmetic" not only reflects his summary and innovation of the traditional mathematical system, but also is a historical witness to the transformation of Chinese arithmetic to modern mathematics.
"Zegu Xi Zhai Arithmetic"
Zegu Xi Zhai Arithmetic is a collection of Li Shanlan's treatises, including thirteen mathematical works. Among them, there are not only traditional Chinese arithmetic works such as "Four-Element Solution", "Stacking Ratio", "Arc Arrow Revelation", but also works that effectively imitate Western mathematics and sort out the mathematical knowledge introduced into China before the end of the Qing Dynasty, such as "Fangyuan Interpretation" and "New Oval Technique", as well as works that have studied Western mathematics, such as "Logarithmic Tip Cone Variation Method Interpretation" and "Series Return".
Figure 2 Title page and preface to "Zegu Xi Zhai Arithmetic"
In the first volume of "Fang Yuan's Interpretation", Li Shanlan proposed the theory of "sharp cone technique", which was independently created and invented before he was exposed to Western calculus theory. He summarized the theory of "sharp cone" into ten "know-it-alls", namely:
First, "When you know that the so-called points, lines, and surfaces of the Westerners cannot be without a body." "The point, the small and the small, the line, the long and thin; the face, the wide and thin." Second, "When the body of knowledge can become a surface, the surface can become a line." "The book of the ruler is obtained by folding paper; the silk of the ruler is made of silk." Third, "When knowing the principle of the wired, surface, and body circulation of the multiplicators." "The square is long because of it, the long is the plaque, and the plaque is compound because of it." 4. "Know that all the multipliers can become faces, and all can become lines." V. "When you know the peace, the shape of a cone with a sharp tip." 6. "Know that all the multipliers have spikes." "The bottom of the three-fold pointed cone is square, but the upper four sides are not flat and concave. The more you multiply, the more concave it becomes." 7. "Know that the cones of the apex have the principle of accumulation." VIII. "When the algorithm of the cusp cone is known: take the high multiplication bottom as the real, the multiplication number plus one as the method, divide it, and obtain the cusp cone product." 9. "When it is known that the faces stacked above the two-fold cone can be changed into lines." 10. "When it is known that the cone is flat, it may be a cone of the tip.".
The concept of "sharp cone" created by Li Shanlan is a geometric model that deals with algebraic problems. Modern scholars believe that his description of the "sharp cone curve" is essentially equivalent to giving equations such as straight lines, parabolas, and cubic parabolas. The "sharp conic product" he created is equivalent to the definite integral formula of the power function and the law of item-by-item integration. Li Shanlan also made creative achievements in the expansion of various trigonometric and inverse trigonometric functions, the expansion of logarithmic functions, and the use of calculus methods to deal with mathematical problems.
Figure 3 Excerpts from the ten "Dang Zhi" of "Fang Yuan's Interpretation"
The current copies of the "Zegu Xi Zhai Arithmetic" include the Haining Periodicals of the Fourth to Sixth Years of Qing Tongzhi, the Jinling Periodicals of the Sixth Year of Qing Tongzhi, and the Eight-Year Jiangning Editions of Qing Guangxu. The Documentation and Information Center of the Chinese Academy of Sciences collects the "Zegu Xi Zhai Arithmetic" published in the sixth year of Qing Tongzhi, which includes thirteen kinds of twenty-four volumes of Li Shanlan's mathematical works, and three volumes of conic curve theory. Scholars can make an appointment to request access to ancient books.
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