Mathematical problems on surveying in ancient China (Q1081588)

In 263 AD Liu Hui wrote the Haidao suanjing after he wrote a commentary on the most famous Jiu Zhang suanshu. All the nine problems are on distant measurement. By tradition, proofs and derivations are written in the commentary together with the diagrams. It has been lost. The authors discuss four possible methods of derivations. They base on the conceptualisation of areas of rectangular blocks, the proportionality of similar rectangular triangles and the principle of chu ru xiang bu. The first two methods were known by later Chinese mathematicians such as Qin and Young but not the last two methods which the authors believe are essential for both the invention and solution of the problems. The important conclusions are: first, trigonometry is not involved; second, since the technical terms are brief enough and the language used is both concise and precise, it is quite probable as Mikami remarked that Liu was acquainted with ''algebraical manipulations''; third, the problem that Liu improvised travelled through India, Muslim countries and into Europe.