Liu Hui · Sea Island Mathematical Manual · Problem 1
Check the
second sighting.
Two poles. Two retreats. One island you cannot reach.
The printed commentary names one divisor. A smaller editorial note corrects it. Try both: a calculation can satisfy the first observation and still miss the second.
Work the survey01 / Make a record
Observe first. Calculate second.
Model the source exercise with equal vertical poles on one level baseline and each eye on the ground. No angle is given or measured. Every length below is in bu.
Here 1 bu = 6 chi (尺), 1 zhang (丈) = 10 chi, and 1 li (里) = 300 bu. Thus the source's three-zhang poles are five bu high. These are units in the exercise; no conversion to metres is assumed. The following pine problem converts fifty bu to three hundred chi across openings 12–13.
Both rays are constructed from their own pole and eye, before testing the candidate. Dotted height: their forward intersection, when one exists.
02 / Operate the recipe
Why does checking the second sighting matter?
Put the front pole at x = 0, rear pole at x = s, and the island foot at x = −D. Each pole has height h. The ground eyes are at x = a and x = s + b.
Front line: y = h(a − x) / a
Rear line: y = h(s + b − x) / b
At the common peak: H = h(a + D) / a = h(s + b + D) / b
So D(b − a) = as; H = h + hs/(b − a).
For any nonzero divisor f, the recipe's candidate D = as/f and H = h + hs/f fits the front line exactly. The second line rejects it unless f = b − a. When b = a, the lines are parallel; when b < a, their intersection is behind the front pole and below pole height, outside the forward-island setup.
This coordinate derivation is modern. It explains the model here; it is not presented as Liu Hui's own proof.
03 / Read three voices
A procedure, a gloss, a correction.
Selected Classical Chinese clauses with new English renderings and modern Mandarin pronunciation aids. Punctuation is supplied; the original column layout and small notes remain visible in the whole photographed openings.
Read columns from right to left, top to bottom. Opening numbers count PDF pages. The two photographed leaves remain together, with margins, stamps and all notes.
Which witness is this?
Haidao suanjing, Waseda University Library ni02 00649, catalogued as 1777. The bibliographic preface is dated the fourth month of Qianlong 40 (1775). Its header attributes the work to Liu Hui of Jin and the commentary to Li Chunfeng of Tang. The work is associated with Liu Hui's third-century mathematics; this photographed Qing book is a much later printed witness.
The Qing preface says the material was compiled from the Yongle Encyclopedia. The contradictory wording in the commentary is what this witness transmits; we do not infer where that reading originated. The ten excerpts here are selected clauses, not a complete critical edition or a reconstruction of ancient pronunciation.
Public-domain scan and catalogue description · Waseda catalogue · Ang Tian Se and Frank J. Swetz (1986), historical study
04 / A modern measurement experiment
Same allowance. Different islands.
Could the same ±2 bu allowance leave height fixed, or without any finite upper bound? Change which error pairs are allowed. Pole height and pole gap stay exact.
The source exercise reports exact numbers. This slider is our experiment, rather than an error estimate supplied by the text. It does not include uneven terrain, unequal poles, eye height or atmospheric effects.
Choose one pair. Keep the whole set.
At ±2 bu on the source exercise, take the front to 100% and rear to −100% for parallel sightings. Make the pair equal, then switch the complete assumption to Shared offset.
05 / Keep an inspectable record
Take the observations with you.
Save the applied record and current operation, the exact fractions, both sighting checks, and the chosen uncertainty interval. The reading copy includes all ten passages, five whole source openings, embedded fonts, source identities, and a blank working sheet. It needs no scripts or network.
Offline edition has not been checked.