Huanrong jiaoyi 圜容較義

Huanrong jiaoyi 圜容較義 "The comparative relation of circles and spheres" is a brief book on astronomical geometry written by Li Zhizao 李之藻 (1565-1630), a disciple of the Italian Jesuit Matteo Ricci (1552-1610; Ch. Li Madou 利瑪竇). He also wrote the book Hungai tongxian tushuo 渾蓋通憲圖說.

This text discusses the comparative geometry of graphical relationships (bijiao tuxing guanxi jihexue 比較圖形關系幾何學), covering topics such as polygons (duobianxing 多邊形), the relationships between polygons and circles (yuan 圓), cones (zhuiti 錐體) and prisms (lengzhuti 棱柱體), regular polyhedra (zheng duomianti 正多面體), and the relationship between spheres (hunyuan 渾圓) and regular polyhedra.

The general conclusion is that for polygons with equal perimeters, the regular polygon with equal side lengths (bianchang xiangdeng 邊長相等) always has a larger area than an irregular polygon with unequal side lengths (bianchang bu deng 邊長不等). Among regular polygons, those with more sides always have a larger area than those with fewer sides. Therefore, when the perimeter is fixed, the area of any polygon is always smaller than that of a circle. Similarly, when the surface area is equal, the sphere has the largest volume.

The preface summarises: "One uses the plane to deduce the sphere (jie pingmian yi tui liyuan 借平面以推立圓) and establishes angles to verify the celestial bodies (she jiaoxing yi zheng hunti 設角形以徵渾體)."

This conclusion was first discovered by the Greek mathematician Zenodorus (c. 200-c. 140 BCE), later organized by Pappus of Alexandria (c. 290-c. 350 CE), and further developed in Europe in the early 16th century. Through the compilation and translation efforts of Matteo Ricci and Li Zhizao, this knowledge was introduced to China.

The book was finished in 1608 and first printed in 1614. The text is included in the series Tianxue chuhan 天學初函, Shoushange congshu 守山閣叢書, Haishanxianguan congshu 海山仙館叢書, Saoye Shanfang congchao 掃葉山房叢抄, Congshu jicheng chubian 叢書集成初編 and Zhong-Xi suanxue congshu 中西算學叢書.