Abstract: The method and proof of finding the area of a triangle from its three sides (hereinafter referred to as “tripartite quadrature”) first appeared in the Metrica and Dioptra of Heron in Greece. Similar or identical formulas later emerged in China, India, the Arab world and Western Europe. From the perspective of cross civilizations, this paper studies the tripartite quadrature from four aspects: propositional form, proof process, knowledge traceability and civilizational exchange, and puts forward three cultural forms of its development: different civilizations employing different methods—the Greek arithmetical and geometrical methods, the Chinese method of Gougu, and the algebraic and geometric methods in the Arab tradition; within the same civilization, different methods—Heron and Boethius in Greece, as well as Al-Khwarizmi and Banu Musa in the Arab word, used different methods; different civilizations sharing the same methods—India and China, or China and Western Europe, where analogous procedures can be identified. This paper argues that the different methods across civilizations reflect the richness, diversity and autonomy of of mathematical cultures; the same method across civilizations illustrates possible transmission and communication between different civilizations; the different methods within a single civilization demonstrate the independence of the minds of mathematicians. In addition, Ming- and Qing-dynasty records and adaptations of “Heron’s formula” show the localization of western mathematical knowledge in the process of dissemination and integration in China.

Key Words: Cross civilizations; Tripartite quadrature; Greece; Arab; Western Europe