Liu Hui
Native spelling: 劉徽
Chinese mathematician of the Cao Wei state whose 263 CE commentary on The Nine Chapters on the Mathematical Art turned a manual of computational recipes into a text with logical justifications, including a dissection-based proof of the Pythagorean (gou-gu) theorem and a polygon method that approximated pi to four correct decimal digits.
Very little is recorded about Liu Hui's personal life; he is thought to have been a descendant of the marquis of Zixiang of the Eastern Han dynasty and to have lived and worked in the state of Cao Wei during China's Three Kingdoms period, in the region of present-day China. No verified birth or death year survives in the sources consulted for this page, so only his best-documented act of authorship is dated here.
In 263 CE Liu Hui completed a commentary on The Nine Chapters on the Mathematical Art (Jiuzhang Suanshu), the classic compilation of problems and procedures that had served Chinese mathematics for centuries, much as Euclid's Elements did in the Greek world. Where the original Nine Chapters gave algorithms without justification, Liu Hui supplied proofs: he gave a dissection proof of the gou-gu (right-triangle) theorem equivalent to the Pythagorean theorem, clarified the logical basis of the array method for solving systems of linear equations (an ancestor of Gaussian elimination) including the use of positive and negative counting rods, and investigated the volumes of pyramids, wedges and other solids by cutting them into simpler pieces — an approach in the spirit of the later Cavalieri principle.
Liu Hui's most celebrated technique is his circle-splitting algorithm for pi: by repeatedly doubling the number of sides of polygons inscribed in a circle, starting from a hexagon and reaching a 3072-gon, he obtained the approximation pi ≈ 3.1416, correct to four decimal places, together with a clear argument for why the method converges. The Sea Island Mathematical Manual (Haidao Suanjing), traditionally attributed to him and later circulated separately, used similar-triangle methods to measure the heights and distances of inaccessible objects such as mountains and islands.
Workplaces: Cao Wei state, Three Kingdoms period, China
China