The Nine Chapters on the Mathematical Art, Vols. 1-3

Imperial Edition of the Siku Quanshu

...are few. Even if there are many scholars and men of wide learning in the world, they may not necessarily be able to master this. The Rites of Zhou the section on the Minister of Instruction states: "On the day of the Summer Solstice, at noon, set up an eight-foot gnomon a vertical rod used to measure shadows. Its shadow is one foot and five inches." It is said that the distance to the point directly below the sun is 15,000 li. This is stated because it was calculated using mathematical arts. Examining the Nine Chapters, the methods for measuring distance using four gnomons, or for measuring mountains by observing trees, all involve reciprocal observation from the sides; there is no such over-reaching or jumping to conclusions. If this is the case, the methods of Zhang Cang and his peers were still not enough to exhaust all numbers. I have searched the Nine Numbers and found the name Chong Cha Double Difference/Triangulation. Tracing its significance, it is precisely for application in such cases. Whenever one observes extreme heights, measures extreme depths, and simultaneously determines their distance, one must use the Chong Cha. The Gou Gu Pythagorean/Right Triangle theorem must use the Chong Cha as its ratio; hence it is called Chong Cha. Set up two gnomons in the city of Luoyang, each eight feet high, facing North-South.