About Yi JingExamples of Yi and Chinese Sciences
Yi Gauges Using Stone and Jade Knives
The previous archaeological studies of stone and jade knifes with holes have found that a complete set of stone knifes with odd ordinal numbers of holes from 1 to 13, which was uncovered at the third phase remains of Xuejiagang Culture in Xuejiagang site, Anhui Province of China, was typologically and chronologically the most complete form of the prehistoric knife-like artifact culture and of a profound calendrical knowledge background. Based on the archaeological evidences in consistent with the ancient documentary records, the present paper proves the following concludes.
As the calendrical knowledge background of stone and jade knifes with holes, the Sifen (four quadrantal) Calendar in which the tie between solstitial points is the ordinate axis of the solar year coordinate and the tie between equinoctial points is its abscissae axis was spread in the form of Fuxi's Bagua (the Eight Trigrams) over the Yellow, Yangtze and Liao river valleys during the Neolithic Age.
The four cardinal points of a solar year were determined with the longest, the shortest and their average length of sun-shadow that were found by frequently measuring the sun-shadow lengths all year round in these regions.
According to the ordinal numbers deployed by the Fuxi's Bagua - 1, 2, 4, 8, 24, the implementing system of the Sifen (four quadrantal) Calendar experienced a long-historic process of the development of sun-shadow length measurement focus from one and two solstitial points to four cardinal points, eight periods' points through twenty-four solar terms' points. Correspondingly, the number of holes made on stone knives increased from 1 or 2 to 3, 5, 7 through 13 before the third phase of Xuejiagang Culture.
The complete set of stone knifes were the gauges for measuring the sun-shadow lengths for each of the twenty-four solar terms and the highest form of implementing the Sifen (four quadrantal) Calendar by using stone and jade knives with holes as the sun-shadow measuring gauges.
With the transition from the Neolithic Age to the Bronze Age, the Sifen (four- quadrantal) Calendar was transformed into the integrated Sifen (four quadrantal) - Wuxing (five phase) Calendar in which the cardinal points were fixed by observations of the sun, the moon and the stars. Correspondingly, the number of holes made on stone and jade knives decreased to 5, 3 through 2 or 1 and then all stone and jade knives with holes were finally replaced by the jade articles named "Yazhang".
This paper verifies these conclusions with the prehistory of Chinese calendrical science and the evolution of sun-shadow measurement instruments in prehistoric China shown in evidences. In the end, it explains the academic significance of the discovery of the truth of stone and jade knifes with holes for exploring the origin and the prehistoric development of the Chinese civilization and science.
What is the Earliest Chinese Origin of ‘Pascal’ Triangle
by Deng Honghai
Since the 17th century, the triangle of binomial coefficients has been known in Europe as ‘Pascal’ Triangle. Having studied the several mathematical texts published in the Song Dynasty, Dr. Joseph Needham, the author of Science and Civilisation in China, used the book’s section titled “Binomial Theorem and ‘Pascal’ Triangle” to introduce the ancient Chinese mathematicians’ contributions to this field. He wrote: “Pascal would have been rather surprised if they could ever have seen Chu Shih-Chieh’s Ssu Yuan Yu Chien (朱世傑﹕四元玉鑑 ) of +1303, from which the triangle of binomial coefficients is here reproduce in Fig.80. It is called the ku fa chhi chheng fang thu (古法七桀方圖 , Old method Chart of the Seven Multiplying Squares). …The fact that Chu speaks of the triangle as old or ancient suggests the binomial theorem had already been understood at least by the beginning of the +12th century.
“According to his finding of the earliest extent Chinese representation of the triangle from Yang Hui’s Hsiang Chieh Chiu Chan Fa (楊輝的詳解九章算法 ) of +1261, Dr. Joseph Needham pointed out that the triangle had existed long before. Chia Hsien (賈憲 ) expounded it about +1100. The method used by Chia (賈憲 ) was called ‘Li-chheng shih so’ (立成釋鎖, the tabulation system for unlocking binomial coefficients). Yang Hui took the triangle from an earlier book, Shih So Suan Shu (釋鎖弄書). The Yung-Lo Ta Tien of the Ming Dynasty (+1407) recorded: “ Yang Hui systematically analyzed the origin of the method for extracting square roots, based on Shih So Suan Shu (釋鎖弄書). Chia Hsien had used this method.”
As shown in Fig.80, the ancient Chinese practice in extracting square or cubic roots was completed by using counting rods. As Dr. Joseph Needham pointed out: “In Fig. 80 it will be noticed that the counting-rod numerals are so turned as to presuppose that the bottom of the triangle originally stood vertically on the left. Thus the powers of the unknown would have stood on successive horizontal rows of the counting –board, exactly as we know (from the Chiu Chang) was the ancient (Han) practice in in extracting square or cubic roots. Here again we can see the logical continuity between the ancient counting – board and the notation of the Sung algebra.” (Joseph Needham: Science and Civilisation in China, Cambridge at the University Press, 1959, P. 133-141).
All the texts listed above have confirmed two conclusions: (1) the triangle had existed long before in China; (2) the triangle had come from the ancient Chinese mathematics of numbering by counting-rods. However, how long the triangle had existed before in China? How and why ancient Chinese cultures could give birth to it? These are remaining questions.
