Haidao Suanjing

Haidao Suanjing (海岛算经; The Sea Island Mathematical Manual) was written by the Chinese mathematician Liu Hui of the Three Kingdoms era (220–280) as an extension of chapter 9 of The Nine Chapters on the Mathematical Art. During the Tang Dynasty, this appendix was taken out from The Nine Chapters on the Mathematical Art as a separate book, titled Haidao suanjing (Sea Island Mathematical Manual), named after problem No 1 "Looking at a sea island." In the time of the early Tang dynasty, Haidao Suanjing was selected into one of The Ten Computational Canons as the official mathematical texts for imperial examinations in mathematics.

This book contained many practical problems of surveying using geometry. This work provided detailed instructions on how to measure distances and heights with tall surveyor's poles and horizontal bars fixed at right angles to them. The unit of measurement was 1 li = 180 zhang= 1800chi, 1 zhang = 10 chi, 1 chi = 10 cun, 1 step(bu) = 6 chi. Calculation was carried out with place value decimal Rod calculus.

Liu Hui used his rectangle in right angle triangle theorem as the mathematical basis for survey. With his "In-out-complement" principle, he proved that the area of two inscribed rectangles in the two complementary right angle triangles have equal area, thus

Q: Now surveying a sea island, set up two three zhang poles at one thousand steps apart, let the two poles and the island in a straight line. Step back from the front post 123 steps, with eye on ground level, the tip of the pole is on a straight line with the peak of island. Step back 127 steps from the rear pole, eye on ground level also aligns with the tip of pole and tip of island. What is the height of the island, and what is the distance to the pole ?

A: The height of the island is four li and 55 steps, and it is 120 li and 50 steps from the pole.

Algorithm: Let the numerator equals to the height of pole multiplied by the separation of poles, let denominator be the difference of ofsets, add the quotient to the height of pole to obtain the height of island.

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