Solve problem 3 of the sea island mathematical manual

Move away bu from the front pole and observe the peak of the island from ground level; it is seen that the tip of the front pole coincides with the peak. Move backward bu from the rear pole and observe the peak of the island from ground level again; the tip of the back pole also coincides with the peak. The diagram below shows the basic set-up of the Sea Island problem. Lines that appear parallel are parallel.

Point A is the top of a mountain on an island out at sea. Point C is at sea-level, directly below A. A surveyor on the mainland DG wishes to determine the height of the mountain CA. However, she cannot get to any of the points A, B or C. What is AC? The following questions are intended as scaffolding to take you deeper into this problem.

Both poles are perpendicular to the ground and they both have a height of 3 Zhang; the distance of the two poles is steps. Similarly, one would go steps backward from the second pole in order to see from the ground that the top of the pole and the peak of the sea island to coincide to the same point. Liu Hui attached his answer and method to the question saying that: The height of the sea island is 4Li and 55 steps; the distance from the sea island to the first pole is Li and steps.

He explained that if we multiply the height of the pole by the distance between the two poles; divide the number we get by the difference of the ofsets steps — steps , then plus this number together with the height of the pole could we get the height of the sea island. For the distance between the closer pole to the island, multiply the ofset of the first pole by the distance of the two poles; divide this number again by the difference of ofsets; the final number we get is the distance between the island and the first pole.

Geometrically, we could illustrate the problem using following figure:. Thus we have that. With this, we could solve further for PQ using our algorithm above. If we plug equation 1 to either of the PQ equation, we would get. Therefore, Liu Hui was completely correct on this question.

The complicity of the questions has been noticed by many western scholars so far, even though the problems are likely to be in the same type. I will discuss more into question 9 in my next blog. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account.

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