Quick Answer: How Did Liu Hui Measure an Island From Shore?
Liu Hui (刘徽, Liú Huī) used two equal poles and two sight lines to work out the height and distance of an island peak. His Sea Island Mathematical Manual (海岛算经, Hǎidǎo Suànjīng) illustrates the method called double difference (重差, chóngchā).[1][2][3][4]
- Known quantities: Both poles have the same height, and the spacing between them is known.[1][3]
- The decisive difference: The observer steps back 123 bu behind the front pole and 127 bu behind the rear one to line up each pole top with the peak. The two setbacks differ by 4 bu.[1][2]
- The results: The peak is 1,255 bu high and 30,750 bu from the front pole in the model described.[1][2][3]
The calculation uses distances available on shore. It assumes the poles and sightings follow the geometry specified in the problem.[1][3]
Suppose you can see the top of an island but cannot walk across the water to measure it. Aligning the peak with one pole gives you an angle, and many different heights and distances could fit that view. A second sighting from a known position supplies the information the first one lacks.[2][3]
Why One Sighting Is Insufficient
In the manual, the two upright poles stand on the same line toward the island. At ground level behind the front pole, the observer finds a position from which the top of that pole lines up with the summit. An equivalent sighting uses the rear pole. The geometry treats the ground under the poles and the base of the island as one level.[1][2][3]
Each sighting produces a small right triangle whose vertical side is the pole and whose horizontal side is the observer’s setback. A larger triangle has the same slope and extends to the summit. Within each sighting, the small and large triangles share the sight angle, so their corresponding sides have the same ratio. This is a different use of right triangles from applying the gougu theorem to three sides of a single triangle.[2][3][4]
What the Double Difference Measures
Write h for the pole height, d for the spacing between the poles, and a and b for the setbacks behind the front and rear poles. With a and b measured from their own poles, the useful difference is b − a. In the example it is 127 − 123, or 4 bu.[1][2][3]
Let H be the summit height above the shared ground level. Compare the portions of the two sight lines beyond the pole tops. Each rises by H − h to the summit. The rear sight line covers an additional horizontal distance d. The common rise and this difference give H − h = h × d ÷ (b − a). The horizontal distance D from the front pole to the island is a × d ÷ (b − a).[1][2][3]
Adding h to the first result restores the height from ground level. The second result already starts at the front pole. Measuring either quantity from a different point would answer a different question.[1][2]
Working the Island Problem in Historical Units
The manual gives the pole height as 3 zhang, or 5 bu in the conversion used for this problem. The spacing between poles is 1,000 bu. Divide by the difference of 4 bu. The height above the pole tops is 5 × 1,000 ÷ 4 = 1,250 bu. Add the height of a pole to get 1,255 bu.[1][2][3]
For the distance, the front setback of 123 bu supplies the other input. Multiply 123 by 1,000 and divide by 4 to obtain 30,750 bu. With 300 bu to a li in the text’s arithmetic, the answers are 4 li 55 bu for the height and 102 li 150 bu for the distance. These are historical units carried through the calculation.[1][3]
Where the Sea Island Manual Fits
The manual is associated with Liu Hui’s commentary on the right-triangle chapter of The Nine Chapters on the Mathematical Art. It contains nine problems involving measurements that observers cannot take directly. The island example supplies a particularly clear case because the unequal setbacks are given and both unknowns can be checked arithmetically.[2][4]
The comparison with Liu Hui’s other mathematical work shows how one author used right triangles for different purposes. In his calculation of pi, a right triangle is constructed inside a geometric figure. Here the two triangles arise from observations made at known positions.[2][4]
FAQs About Chinese Surveying Mathematics
Is this the same as the Pythagorean theorem?
Both topics use right triangles. The island solution relates the sizes of triangles with matching angles, which lets the observer use two sight lines to find the unknown quantities.[2][3]
Where is the island’s distance measured from?
The distance of 30,750 bu starts at the front pole. The front observer stands another 123 bu behind it.[1][2]
Does the text identify a real island?
The passage presents a mathematical problem about viewing an island. It supplies the geometry and measurements for the exercise, without naming a place.[1][4]
Sources
[1] Chinese Text Project. Hai Dao Suan Jing, the Sea Island Problem
[2] MacTutor History of Mathematics. Liu Hui
[3] Louisiana State University. The Sea Island Problem
[4] Oxford Academic. Sea Island Mathematical Manual