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Is the Fuxi liushisi gua fangwei (伏羲六十四卦方位) diagram attributed to Shao Yong binary? Clarifying a consequence of its analogy with the binary arithmetic of Leibniz
Published online by Cambridge University Press: 15 August 2025
Argument
The Jesuit Joachim Bouvet established an analogy between the binary arithmetic developed by Leibniz and the diagram Fuxi liushisi gua fangwei (or FX64), attributed to Shao Yong, which organizes the sixty-four hexagrams according to the Fuxi/Xiantian order. Consequently, this diagram could be considered as binary. Some scholars argue that the diagram is not binary because of the different construction of the two systems and the “wrong” reading direction used by Bouvet and Leibniz—opposite to the one used in China. Nevertheless, by a superimposition of Leibniz’s binary table and of the derivation table used to construct the diagram, this article shows that the diagram is binary, since it is constituted of two elements and the binary system can use other symbols than 0 and 1. The reverse methodology used in constructing the two systems because of their different purpose—division for the FX64 diagram and multiplication for Leibniz’s dyad—allows their reading from either one direction or the reverse. This does not affect the fact that they are both binary, since it leads to the same form and structure.
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References
References
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Cammann, Schuyler. 1991. “Chinese Hexagrams, Trigrams, and the Binary System.” Proceedings of the American Philosophical Society 135 (4): 576–589.Google Scholar
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Mungello, David E. 1977. Leibniz and Confucianism: The Search for Accord. Honolulu: University Press of Hawaii.Google Scholar
Mungello, David E. 1989. Curious Land, Jesuit Accommodation and the Origins of Sinology. Honolulu: University Press of Hawaï.Google Scholar
Needham, Joseph. 1956. Science and Civilisation in China. Vol. 3. Cambridge: Cambridge University Press.Google Scholar
Nikolic, Aleksandar. 1994. “Gottfried Wilhem Leibniz et le système binaire.” Review of Research, Faculty of Sciences, Mathematics Series (University of Novi Sad) 24 (2): 69–87.Google Scholar
Perkins, Franklin. 2004. Leibniz and China: A Commerce of Light. Cambridge University Press.Google Scholar
Ryan, James A. 1996. “Leibniz’ Binary System and Shao Yong’s ‘Yijing’.” Philosophy East and West 46 (1): 59–90.10.2307/1399337CrossRefGoogle Scholar
Schöter, Andreas. 1998. “Boolean Algebra and the Yi Jing.” The Oracle: The Journal of Yijing Studies 2 (7): 19–34.Google Scholar
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Leibniz, Gottfried Wilhelm. 1703. “Explication de l’Arithmétique binaire, qui se sert des seuls caractères 0 et 1, avec des remarques sur son utilité, et sur ce qu’elle donne le sens des anciennes figures Chinoises de Fuxi”, Mémoires de mathématique et de physique de l’Académie royale des sciences, Académie royale des sciences. https://hal.archives-ouvertes.fr/ads-00104781
Google Scholar
Leibniz, Gottfried Wilhelm. 1875–90. Die mathematische Schriften (I-VII). Edited by Gerhardt, C. I.. Berlin: Weidmann.Google Scholar
Leibniz, Gottfried Wilhelm. 1923. Sämtliche Schriften und Briefe. Edited by Académie des Sciences de Berlin, Series I–VII. Darmstadt, Leipzig, and Berlin : De Gruyter.Google Scholar
Shao, Yong (邵雍). “Huangji jingshi shujie” (皇極經世書解) [Interpretation of the Book of the August Ultimate through the Ages], in Qinding siku quanshu (欽定四庫全書) [Complete books of the Four Storehouses], Edited by Wang Zhi (王植). Juanshou shang/xia 卷首上/下 [Parts one and two].Google Scholar
Shao, Yong 邵雍. 1988. “Huangji jingshi”皇極經世 [Book of the August Ultimate through the Ages], in Daozang 道藏 [Daoist Canon], Shanghai:上海市: Shanghai Bookstore Publishing House上海书店出版社.Google Scholar
Shao, Yong 邵雍. 2004. Huangji jingshi shujin shuo. Guanwu neipian/yanxiu zhuanjishuo 皇極經世書今說 [Contemporary Interpretation of the Book of the August Ultimate through the Ages]。觀物內篇/閆修篆輯說. Taipei shi: Lao gu wenhua shiye gongsi.Google Scholar
Widmaier, Rita and Babin, Malte-Ludolf (Editors). 2006. Der Briefwechsel mit den Jesuiten in China (1689-1714). Hamburg: Felix Meiner Verlag.Google Scholar
Zacher, Hans J. 1973. Die Hauptschriften zur Dyadik von G. W. Leibniz. Ein Betrag zur Geschichte des binären Zahlensystems. Frankfurt am Main: Vittorio Klostermann.Google Scholar
Zhouyi《周易》[The Book of Changes], https://ctext.org/book-of-changes/zh.Google Scholar
Zhu, Xi 朱熹. 2004. Zhouyi benyi 周易本義 [Original Meaning of the Zhouyi]. Taipei 臺北市: Daan Press 大安出版社.Google Scholar
Aiton, Eric J., and Shimao, Eikoh. 1981. “Gorai Kinzo’s Study of Leibniz and the I Ching Hexagrams.” Annals of Science 38: 71–92.CrossRefGoogle Scholar
Alcantara, Jean-Pascal. 2006. ““Cette caractéristique secrète et sacrée…”: Leibniz et Bouvet, lecteurs du Yijing.” Intervention au séminaire du Centre d’Études et de Recherches sur l’Humanisme et l’Âge Classique (UPRES-A CNRS 5037) Chiffres et secrets, organisé par M. Dominique Descotes, 21 janvier 2006, Université de Lyon III.Google Scholar
Arrault, Alain. 2000. “Les diagrammes de Shao Yong (1012–1077). Qui les as vus ?” Etudes chinoises XIX (1–2): 67–114.10.3406/etchi.2000.1289CrossRefGoogle Scholar
Arrault, Alain. 2001. “Mutations et calcul binaire de la Chine à l’Europe.” Recherches Sociologiques 3: 81–96.Google Scholar
Arrault, Alain. 2002. Shao Yong (1012-1077), poète et cosmologue. Paris: Collège de France, IHEC.Google Scholar
Arrault, Alain. 2013. “Numbers, Models and Sounds: Numerical Speculations of Shao Yong (1012–1077).” Monumenta Serica 61: 183–201.CrossRefGoogle Scholar
Bae, Sun Bok. 2001. “Die Leibniz binäre Arithmetik und das I-Ching Symbolik der Hexagramme vom Standpunkt der modernen Logik.” [Leibniz binary arithmetic and the I-Ching symbolism of the hexagrams from the point of view of modern logic] Nonliyeongu 논리연구 [Korean journal of logic] 5 (1): 147–157.Google Scholar
Birdwhistell, Anne D. 1989. Transition to Neo-Confucianism: Shao Yung on Knowledge and Symbols of Reality. Stanford California: Stanford University Press.Google Scholar
Brach, Jean-Pierre. 1994. La symbolique des nombres. Paris: Presses Universitaires de France, “Que sais-je?”Google Scholar
Cammann, Schuyler. 1991. “Chinese Hexagrams, Trigrams, and the Binary System.” Proceedings of the American Philosophical Society 135 (4): 576–589.Google Scholar
Ching, Julia, and Oxtoby, Willard G.. 1992. “Moral Enlightenment, Leibniz and Wolff on China.” Monumenta Serica: Monograph Series XXVI. Nettetal: Steyler Verlag.Google Scholar
Code GROS-GRAY. Edited by Gérard Villemin. April 2017. http://villemin.gerard.free.fr/Wwwgvmm/Numerati/CodeGray.htm (accessed 13 May 2021).Google Scholar
Gu, Mingdong. 2009. “The Theory of the Dao and Taiji: A Chinese Model of the Mind.” Journal of Chinese Philosophy 36 (1):157–175.CrossRefGoogle Scholar
Han, Qi. 2015. “Les ouvrages compilés et imprimés au Palais sous Kangxi.” In Imprimer sans profit? Le livre non commercial dans la Chine impériale. Edited by Bussotti, M. and Drège, J.-P., 531–552. Geneve: Librairie Droz.Google Scholar
Lalande, André. 1999. Vocabulaire technique et critique de la philosophie. Vol. 1 and 2. Paris: Presses Universitaires de France.Google Scholar
Makeham, John. 1986. “The History of the Development of Zhou Yi (周易) Studies in the West: An Overview.” Zhongguo yanjiu jikan (中国研究集刊) [Journal of Chinese Studies] 3: 40–57.Google Scholar
Martzloff, Jean-Claude. 2009. Le calendrier chinois. Structure et calculs (104 av. J.-C. – 1644). Paris: Honoré Champion.Google Scholar
McKenna, Stephen E. and Mair, Victor H.. 1979. “A Reordering of the Hexa-Grams of the I Ching.” Philosophy East and West 29 (4): 421–441.10.2307/1398813CrossRefGoogle Scholar
Mungello, David E. 1977. Leibniz and Confucianism: The Search for Accord. Honolulu: University Press of Hawaii.Google Scholar
Mungello, David E. 1989. Curious Land, Jesuit Accommodation and the Origins of Sinology. Honolulu: University Press of Hawaï.Google Scholar
Needham, Joseph. 1956. Science and Civilisation in China. Vol. 3. Cambridge: Cambridge University Press.Google Scholar
Nikolic, Aleksandar. 1994. “Gottfried Wilhem Leibniz et le système binaire.” Review of Research, Faculty of Sciences, Mathematics Series (University of Novi Sad) 24 (2): 69–87.Google Scholar
Perkins, Franklin. 2004. Leibniz and China: A Commerce of Light. Cambridge University Press.Google Scholar
Ryan, James A. 1996. “Leibniz’ Binary System and Shao Yong’s ‘Yijing’.” Philosophy East and West 46 (1): 59–90.10.2307/1399337CrossRefGoogle Scholar
Schöter, Andreas. 1998. “Boolean Algebra and the Yi Jing.” The Oracle: The Journal of Yijing Studies 2 (7): 19–34.Google Scholar
Secter, Mondo. 1998. “The Yin-Yang System of Ancient China: The Yijing-Book of Changes as a Pragmatic Metaphor for Change Theory.” Paideusis - Journal for Interdisciplinary and Cross-Cultural Studies 1: 85–106.Google Scholar
Shi, Zhonglian. 2000. “Leibniz’s Binary System and Shao Yong’s Xiantiantu.” In Das Neueste Über China, G. W. Leibnizens Novissima Sinica von 1697, edited by Wenchao, Li and Poser, Hans, 165–169. Stuttgart: Franz Steiner Verlag.Google Scholar
Sypniewski, Bernard Paul. 2000. “China and Universals: Leibniz, Binary Mathematics, and the Yijing Hexagrams.” Monumenta Serica: Journal of Oriental Studies 53: 287–314.10.1179/mon.2005.53.1.009CrossRefGoogle Scholar
Wei, Deming (卫德明) and Wang, Xipeng (王汐朋). 2014. “Yijing de shijian guannian” (《易经》的时间观念) [The Concept of Time in the Book of Changes]. Xiandai zhexue (现代哲学) [Modern Philosophy] 3 (134): 85–92.Google Scholar
Wilhelm, Richard. 1950, 2016. Yi king: Le livre des transformations. Translated by Perrot, Etienne. Paris: Médicis.Google Scholar
Wilhelm, Hellmut. 1960. Change: Eight Lectures on the I Ching. New York: Pantheon Books.Google Scholar
Zhang, Yuanshan (张远山). 2016. “Fuxi gua xu tansuo shi” (伏羲卦序探索史) [Exploring the Fuxi Hexagram Sequence History]. Shehui kexue luntan (社会科学论坛) [Social Science Forum]: 22–58.Google Scholar
Zhongguo, Zhao (赵中国). 2008. “Dui ‘xiantian tu yu erjinzhi guanxi’ zhi zheng de kaocha yu bianxi” (对“先天图与二进制关系”之争的考察与辨析) [Investigation and Analysis of the Controversy of the ‘Xiantian Diagram and Binary Relation’]. Zhouyi yanjiu (周易研究) [Zhouyi Research] 1 (87): 75–82.Google Scholar
Qibing, Zhou (周琪兵). 2012. “Bagua tu zhong de shuzi guilu tanjiu” (八卦图中的数字规律探究) [Research on the Law of Numbers in the Eight Diagrams]. Keyan fazhan (科研发展) [Scientific Research Development] 10: 96–97.Google Scholar