Chiu chang suan shu

The Chiu chang suan shu is the most famous Chinese mathematical text. Its content is a summary of the mathematical knowledge possessed in China up to the middle of the 3rd century. This book dates from the period of the Han Dynasty (206 B.C. to A.D. 220). It is an anonymous collection of 246 problems. A book obviously not for beginners as a lot of assumptions are made, that is the basic arithmetical processes are supposed to be known. Each question has 'stated their numerical answer and corresponding arithmetic algorithm' (Gratten-Guinness, 1994, p94). From the third century on there were numerous commentators of the book, such as Liu Hui (A.D. 263), Zu Kengzhi (early sixth century), Li Chunfeng (602-670) and Yang Hui (1270).

It consists of nine distinct chapters on mathematics, 'consisting mainly of a set of problems with general rules for their solution' (Schaefer, 1979, p9). 'Three involve surveying and engineering formulas, three are devoted to problems of taxation and bureaucratic administration, and the remaining three to specific computational techniques' (Swetz and Kao, 1977, p17). Briefly the contents are as follows :

1. Fang thien (land surveying)
   • correct rules for area of a triangle, rectangle, circle and trapezoid
   • pi is taken as 3
   • area of the segment of a circle = (S/2)(C+S), where
     • C = length of a chord
     • S = length of sagitta of the segment
   • addition, subtraction, multiplication and division of fractions

2. Su mi (millet and rice)
   • simple questions on percentage and proportions
   • originally contained a table which converted weights of varied grains in terms of rice

3. Tshui fen (distribution by progression)
   • problems involving taxation, partnership and arithmetric and geometrical progressions
   • uses proportions and the rule of three to solve

4. Shoa kuang (diminishing breadths)
   • finding the sides of figures when areas and single sides are known
   • square and cube roots

5. Shang kung (consultations of engineering works)
   • volumes of solid figures (prisms, cylinders, pyramids, etcetera)

6. Chun shu (important taxation)
   • distribution of taxes and the various difficulties in transporting taxes

7. Ying pu tsu (excess and deficiency)
   • the 'rule of false position'

8. Fang ch'eng (the way of calculating by tabulation)
   • solution of systems of simultaneous linear equations

9. Kou ku (right angles)
   • properties of right angled triangles

Clearly the Chiu chang suan shu gives evidence of the high level of the Chinese insight and computation skills utilised in early China.