Charlotte-V Pollet, The Empty and the Full: Li Ye and the Way of Mathematics. Geometrical Procedures by Sections of Areas

The title of this book reminds me of a dish on the menu in one of my favorite restaurants: the “I cannot make up my mind plate.” It is a response to the problem of decision-making in the face of hunger and too many options to choose from. The author here has a similar problem: confronted with an interest in a large variety of aspects, no clear idea of which direction to head in emerges, and many imprecisions remain in the face of a self-made combinatorial chaos. By the end, the author has cooked up a book that “represents an intersection of the history of mathematics, a description of meditative techniques as described by cultural historians, and the philosophy of language” (xii). The object of study is, put simply, a Chinese mathematical book written in 1259 and published in 1282, the Yigu yanduan 益古演段 (translated as “Development of Pieces [of Areas] [according to] [the collection] Augmenting the Ancient [knowledge]”) by the Yuan dynasty mathematician Li Ye 李冶. Being a collection of sixty-four problems, accompanied with answers, solution procedures (both algebraic and geometrical), and many diagrams, it is the oldest extant text to use the Chinese algebraic method of the so-called “celestial unknown” (tian yuan 天元) to solve problems of the second degree, where also negative coefficients are admitted. All problems concern a kind of configuration of a square and a circular field, inscribed one into the other or intersecting each other in various ways. The basic assumption that the author makes is that Li Ye’s book—like any other—is untranslatable, that meaning lies not in discourse but in the non-discursive parts: the void between the lines, the structural organization of the book, and the two kinds of visual representations contained in it: one for the givens and the other for the coefficients of the polynomial corresponding to geometrical “pieces of areas.” In spite of this philosophical stance on untranslatability, the present study nevertheless does provide translations (literal and modern mathematical translations) of some