"[The] Sea Island Computational Classic" 《海島算經》

《海島算經》 "[The] Sea Island Computational Classic"

《海島算經》, "[The] Sea Island Computational Classic", is one of the ten canon classics of Chinese mathematics. Written by Liu Huei of the Three Kingdoms period (220–280 AD), it consists of nine problems in the surveying of distant structures or landforms. For each problem, the Classic itself only gives the statement of the problem, the answer and the formula used to compute the answer. Derivations of the formulae we find in 《海島算經細草圖說》, "[The] Sea Island Computational Classic [with] detailed working, diagrams, [and] explanations". Written by Lee Huang, the rough working (草), diagrams (圖) and explanations (說) supplement the older (but at times erroneous) annotations of Lee Chʻun‑fêng. It is in the explanations (說) that we find derivations of the formulae, not unlike high school geometry proofs.

Here I translate the the eponymous first problem 望海島, "looking at [a] sea island", along with its corresponding derivation as found in 《海島算經細草圖說》.

Translation

The Chinese source texts 今有望海島 (for the problem) and 說曰 (for the derivation) are from archive.org. After reading these in their original forms (i.e. without any punctuation), it is hard not to appreciate modern mathematical notation.

Chinese units of measurement have changed many times throughout history. There are three length units used in this excerpt, and while their exact size is not terribly important, it is good to know the order of magnitude:

A chang (丈), "rod", is of the order three metres,

A pu (步), "pace", is of the order two metres, and

A lee (里), "mile", is of the order half a kilometre.

The relevant conversions are

\begin{aligned} 3 \unit{rods} &= 5 \unit{paces}, \\ 1 \unit{mile} &= 300 \unit{paces}. \end{aligned}

The inserted olden-style punctuation is my own. Parentheticals are (half-column) inline annotations in the original text.

Problem

Source text	Target text	Notes
今有望海島、立兩表、齊高三丈、前後相去千步、令後表與前表參相直。	Suppose there be [a] looking at [a] sea island; erect two posts, of equal height three rods, front [and] rear separated from each other [by a] thousand paces, [and] make [the] rear post and [the] front post aligned [the] three with each other.	表： posts; or pillars 齊： equal; or uniform 丈： rod; or measure A unit of distance, of the order three metres. Note that 3 \unit{rods} = 5 \unit{paces}. 步： paces; or steps A unit of distance, of the order two metres. 直： aligned; lit. straight
從前表卻行一百二十三步、人目著地取望島峯、與表末參合。	Walking backward one hundred [and] twenty-three paces from [the] front post, man's eye upon [the] ground, taking [a] looking at [the] peak of [the] island, meet [the] three with [the] tip of [the] post.	著： upon; lit. contacting 峯： peak; or summit
從後表卻行一百二十七步、人目着地取望島峯、亦與表末參合。	Walking backward one hundred [and] twenty-seven paces from [the] rear post, man's eye on [the] ground, taking [a] looking at [the] peak of [the] island, meet [the] three also with [the] tip of [the] post.
問島高及去表各幾何。	[We] ask, how much each [be the] island's height and [the] separation from [the] post?
答曰、島高四里五十五步、去表一百二里一百五十步。	Answer saith: [the] island's height [be] four miles [and] fifty-five paces, [and the] separation from [the] post, one hundred [and] two miles [and] one hundred [and] fifty paces.	里： miles A unit of distance, of the order half a kilometre. Note that 1 \unit{mile} = 300 \unit{paces}. 一百二： one hundred [and] two The square-bracketed [and] is supplied, which is noteworthy because: In modern Chinese, you must say 一百零二, "one hundred oh two", to mean 102. If you just say 一百二 or 百二, it means 120; the 二 by default refers to the next significant digit. In Literary Chinese you might see 一百又二 or 一百有二, "one hundred and two", but the "and" is not necessary. Thus 一百二 or 百二 means 102 rather than 120. This is preserved in modern Japanese.
術曰、以表高乘表間爲實、相多爲法、除之。所得加表高、即得島高。	Method saith: use [the] posts' height times [the] posts' separation as [the] dividend, [and the] mutual excess as [the] divisor, [and] divide them. That which resulteth added to [the] posts' height, doth result in [the] island's height.	間： separation 間、去聲、 Cantonese: kaan3, Mandarin: jiàn 相多： mutual excess; or difference It is not immediately obvious, but "mutual excess" refers to the difference between the backward walking distances from the two posts. In modern notation, \text{島高} = \frac{ \text{\colr{表高}} \cdot \text{\colg{表間}} }{ \text{\colb{相多}} } + \text{\colr{表高}} or \begin{aligned} \textq{island height} &= \frac{ \textq{\colr{post height}} \textq{\colg{post separation}} }{ \textq{\colb{mutual excess}} } + \textq{\colr{post height}} \\[\tallspace] &= \frac{ 5 \unit{paces} \cdot 1000 \unit{paces} }{ 127 \unit{paces} - 123 \unit{paces} } + 5 \unit{paces} \\[\tallspace] &= 4 \unit{miles} + 55 \unit{paces}. \end{aligned}
求前表去島遠近者、以前表卻行乘表間爲實、相多爲法、除之、得島去表里數。	[To] seek [the] distance [that the] front post [is] separated from [the] island: use [the] front post's backward walk times [the] posts' separation as [the] dividend, [and the] mutual excess as [the] divisor. Dividing them, resulteth in [the] number of miles [that the] island [is] separated from [the] post.	In modern notation, \text{島去表} = \frac{ \text{\colr{前表卻行}} \cdot \text{\colg{表間}} }{ \text{\colb{相多}} } or \begin{aligned} \textq{island--post separation} &= \frac{ \textq{\colr{front post backward walk}} \textq{\colg{post separation}} }{ \textq{\colb{mutual excess}} } \\[\tallspace] &= \frac{ 123 \unit{paces} \cdot 1000 \unit{paces} }{ 127 \unit{paces} - 123 \unit{paces} } \\[\tallspace] &= 102 \unit{miles} + 150 \unit{paces}. \end{aligned}

Diagram

Source text

Target text

Notes

Diagram representing a sea island and two posts of equal height.

今有望海島、立兩表、齊高三丈、前後相去千步、令後表與前表參相直。

Suppose there be [a] looking at [a] sea island; erect two posts, of equal height three rods, front [and] rear separated from each other [by a] thousand paces, [and] make [the] rear post and [the] front post aligned [the] three with each other.

表： posts; or pillars
齊： equal; or uniform
丈： rod; or measure
A unit of distance, of the order three metres. Note that 3 \unit{rods} = 5 \unit{paces}.
步： paces; or steps
A unit of distance, of the order two metres.
直： aligned; lit. straight

Diagram showing a backward walk from the front post, such that the sea island's peak, the post's tip, and an eye at ground level are collinear.

從前表卻行一百二十三步、人目著地取望島峯、與表末參合。

Walking backward one hundred [and] twenty-three paces from [the] front post, man's eye upon [the] ground, taking [a] looking at [the] peak of [the] island, meet [the] three with [the] tip of [the] post.

著： upon; lit. contacting
峯： peak; or summit

Diagram showing a backward walk from the rear post, such that the sea island's peak, the post's tip, and an eye at ground level are collinear.

從後表卻行一百二十七步、人目着地取望島峯、亦與表末參合。

Walking backward one hundred [and] twenty-seven paces from [the] rear post, man's eye on [the] ground, taking [a] looking at [the] peak of [the] island, meet [the] three also with [the] tip of [the] post.

Diagram showing the sought-after quantities: the sea island's height and the distance from front post.

問島高及去表各幾何。

[We] ask, how much each [be the] island's height and [the] separation from [the] post?

答曰、島高四里五十五步、去表一百二里一百五十步。

Answer saith: [the] island's height [be] four miles [and] fifty-five paces, [and the] separation from [the] post, one hundred [and] two miles [and] one hundred [and] fifty paces.

里： miles
A unit of distance, of the order half a kilometre. Note that 1 \unit{mile} = 300 \unit{paces}.
一百二： one hundred [and] two
The square-bracketed [and] is supplied, which is noteworthy because:
- In modern Chinese, you must say 一百零二, "one hundred oh two", to mean 102. If you just say 一百二 or 百二, it means 120; the 二 by default refers to the next significant digit.
- In Literary Chinese you might see 一百又二 or 一百有二, "one hundred and two", but the "and" is not necessary. Thus 一百二 or 百二 means 102 rather than 120. This is preserved in modern Japanese.

術曰、以表高乘表間爲實、相多爲法、除之。所得加表高、即得島高。

Method saith: use [the] posts' height times [the] posts' separation as [the] dividend, [and the] mutual excess as [the] divisor, [and] divide them. That which resulteth added to [the] posts' height, doth result in [the] island's height.

間： separation
間、去聲、 Cantonese: kaan3, Mandarin: jiàn
相多： mutual excess; or difference
It is not immediately obvious, but "mutual excess" refers to the difference between the backward walking distances from the two posts.
In modern notation,
\text{島高} = \frac{ \text{\colr{表高}} \cdot \text{\colg{表間}} }{ \text{\colb{相多}} } + \text{\colr{表高}}
or
\begin{aligned} \textq{island height} &= \frac{ \textq{\colr{post height}} \textq{\colg{post separation}} }{ \textq{\colb{mutual excess}} } + \textq{\colr{post height}} \\[\tallspace] &= \frac{ 5 \unit{paces} \cdot 1000 \unit{paces} }{ 127 \unit{paces} - 123 \unit{paces} } + 5 \unit{paces} \\[\tallspace] &= 4 \unit{miles} + 55 \unit{paces}. \end{aligned}

求前表去島遠近者、以前表卻行乘表間爲實、相多爲法、除之、得島去表里數。

[To] seek [the] distance [that the] front post [is] separated from [the] island: use [the] front post's backward walk times [the] posts' separation as [the] dividend, [and the] mutual excess as [the] divisor. Dividing them, resulteth in [the] number of miles [that the] island [is] separated from [the] post.

In modern notation,
\text{島去表} = \frac{ \text{\colr{前表卻行}} \cdot \text{\colg{表間}} }{ \text{\colb{相多}} }
or
\begin{aligned} \textq{island--post separation} &= \frac{ \textq{\colr{front post backward walk}} \textq{\colg{post separation}} }{ \textq{\colb{mutual excess}} } \\[\tallspace] &= \frac{ 123 \unit{paces} \cdot 1000 \unit{paces} }{ 127 \unit{paces} - 123 \unit{paces} } \\[\tallspace] &= 102 \unit{miles} + 150 \unit{paces}. \end{aligned}

Derivation

The derivation in 《海島算經細草圖說》, called an explanation (說), comes with a diagram (圖) which I have reproduced here, minus some irrelevant vertical lines. I have mapped the names of the vertices as follows:

Chinese	甲	乙	丙	丁	戊	己	庚	辛	壬	辰
Latin	A	B	C	D	E	F	G	H	I	O

Source text	Target text	Notes
說曰、甲乙爲海島、甲爲島峯。	Explanation saith: AB is [the] sea island; A is [the] peak of [the] island.
丙丁爲前表、戊丁爲前去表、戊爲人目、戊丙甲爲前表望島峯。	CD is [the] front post; ED is [the] front [walk's] separation from [the] post; E is man's eye; ECA is [the] front post's looking at [the] peak of [the] island.
己庚爲後表、辛庚爲後去表、辛爲人目、辛己甲爲後表望島峯。	FG is [the] rear post; HG is [the] rear [walk's] separation from [the] post; H is man's eye; HFA is [the] rear post's looking at [the] peak of [the] island.
與庚戊丁乙平行作己丙辰、則辰乙與丙丁己庚俱等。	Constructing FCO parallel with GEDB, OB [is] equal with both CD [and] FG.	俱： both; lit. all
與丙戊平行作己壬、則壬庚與戊丁等。	Constructing FI parallel with CE, IG [is] equal with ED.
己庚辛形、與甲辰己形同式。	[The] shape FGH, [is] similar with [the] shape AOF.	同式： similar; lit. same form
己庚壬形、與甲辰丙形同式。	[The] shape FGI, [is] similar with [the] shape AOC.
己庚辛形、分己庚壬形、同己庚句、而辛壬爲股較。	[The] shape FGH, [is] split into [the] shape FGI, of same minor-cathetus FG, with HI being [the] major-cathetus chariot-beam.	句： minor-cathetus (Variant of 勾.) The shorter leg of a right-angled triangle. 股： major-cathetus The longer leg of a right-angled triangle. Surveying the sea island from distances much greater than its height, the horizontal lengths will be much greater than the vertical lengths. Thus the relevant horizontal segments are referred to as major-catheti (股), and the relevant vertical segments as minor-catheti (句). 較： chariot-beam 較、入聲、音覺、 Cantonese: kok8, Mandarin: jiào If I understand the entry in 《康熙字典》 correctly, 較 are horizontal beams of a chariot which stick out like horns, above 輢 (the wooden vertical side boards) and above 軾 (which are the horizontal handrails for standing passengers). So 辛壬爲股較 is a literary way of saying "HI sticks out horizontally" (from FGI).
甲辰己形、分甲辰丙形、同甲辰句、而己丙爲股較。	[The] shape AOF, [is] split into [the] shape AOC, of same minor-cathetus AO, with FC being [the] major-cathetus chariot-beam.
以小股較辛壬比小句己庚、若大股較己丙比大句甲辰。	Using [the] small major-cathetus chariot-beam HI [to] compare unto [the] small minor-cathetus FG, [is] as [the] great major-cathetus chariot-beam FC compared unto [the] great minor-cathetus AO.	In modern notation, \begin{aligned} \frac{\colb{HI}}{\colr{FG}} &= \frac{\colg{FC}}{\colv{AO}} \\[\tallspace] \colv{AO} &= \frac{\colr{FG} \cdot \colg{FC}}{\colb{HI}}. \end{aligned}
既得甲辰、加表高辰乙（即丙丁）、得甲乙、爲島高。	Having gotten AO, add [the] post height OB (which is CD), [to] get AB, [which] be [the] height of [the] island.	In modern notation, \begin{aligned} AB &= \colv{AO} + \colr{OB} \\[\tallspace] &= \frac{\colr{FG} \cdot \colg{FC}}{\colb{HI}} + \colr{OB} \\[\tallspace] \textq{island height} &= \frac{ \textq{\colr{post height}} \textq{\colg{post separation}} }{ \textq{\colb{mutual excess}} } + \textq{\colr{post height}}. \end{aligned} Remember \begin{aligned} \textq{\colb{mutual excess}} &= \colb{HI} \\ &= HG - IG \\ &= HG - ED \\ &= \textq{rear post backward walk} - \textq{front post backward walk}. \end{aligned}
小股較辛壬比分股壬庚、若大股較己丙比分股丙辰（即丁乙）、得島去前表之遠。	[The] small major-cathetus chariot-beam HI compared unto [the] split major-cathetus IG, [is] as [the] great major-cathetus chariot-beam FC compared unto [the] split major-cathetus CO (which is DB), resulting in [the] distance [that the] island [is] separated from [the] front post.	In modern notation, \begin{aligned} \frac{\colb{HI}}{\colr{IG}} &= \frac{\colg{FC}}{\colv{CO}} \\[\tallspace] &= \frac{\colg{FC}}{\colv{DB}} \\[\tallspace] \colv{DB} &= \frac{ \colr{IG} \cdot \colg{FC} }{ \colb{HI} } \\[\tallspace] \textq{\colv{island--post separation}} &= \frac{ \textq{\colr{front post backward walk}} \textq{\colg{post separation}} }{ \textq{\colb{mutual excess}} }. \end{aligned}

Diagram

Source text

Target text

Notes

Creating the diagram from scratch: draw a vertical segment AB representing the sea island, and draw an horizontal line out from B representing ground level.

說曰、甲乙爲海島、甲爲島峯。

Explanation saith: AB is [the] sea island; A is [the] peak of [the] island.

Draw a segment CD representing the front post (with D at ground level), then extend AC until it hits ground level at E.

丙丁爲前表、戊丁爲前去表、戊爲人目、戊丙甲爲前表望島峯。

CD is [the] front post; ED is [the] front [walk's] separation from [the] post; E is man's eye; ECA is [the] front post's looking at [the] peak of [the] island.

Draw a segment FG representing the rear post (with G at ground level), then extend AF until it hits ground level at H.

己庚爲後表、辛庚爲後去表、辛爲人目、辛己甲爲後表望島峯。

FG is [the] rear post; HG is [the] rear [walk's] separation from [the] post; H is man's eye; HFA is [the] rear post's looking at [the] peak of [the] island.

與庚戊丁乙平行作己丙辰、則辰乙與丙丁己庚俱等。

Constructing FCO parallel with GEDB, OB [is] equal with both CD [and] FG.

俱： both; lit. all

Construct FI such that it has the same magnitude and direction as CE.

與丙戊平行作己壬、則壬庚與戊丁等。

Constructing FI parallel with CE, IG [is] equal with ED.

己庚辛形、與甲辰己形同式。

[The] shape FGH, [is] similar with [the] shape AOF.

同式： similar; lit. same form

己庚壬形、與甲辰丙形同式。

[The] shape FGI, [is] similar with [the] shape AOC.

己庚辛形、分己庚壬形、同己庚句、而辛壬爲股較。

[The] shape FGH, [is] split into [the] shape FGI, of same minor-cathetus FG, with HI being [the] major-cathetus chariot-beam.

句： minor-cathetus
(Variant of 勾.) The shorter leg of a right-angled triangle.
股： major-cathetus
The longer leg of a right-angled triangle.
Surveying the sea island from distances much greater than its height, the horizontal lengths will be much greater than the vertical lengths. Thus the relevant horizontal segments are referred to as major-catheti (股), and the relevant vertical segments as minor-catheti (句).
較： chariot-beam
較、入聲、音覺、 Cantonese: kok8, Mandarin: jiào
If I understand the entry in 《康熙字典》 correctly, 較 are horizontal beams of a chariot which stick out like horns, above 輢 (the wooden vertical side boards) and above 軾 (which are the horizontal handrails for standing passengers). So 辛壬爲股較 is a literary way of saying "HI sticks out horizontally" (from FGI).

甲辰己形、分甲辰丙形、同甲辰句、而己丙爲股較。

[The] shape AOF, [is] split into [the] shape AOC, of same minor-cathetus AO, with FC being [the] major-cathetus chariot-beam.

以小股較辛壬比小句己庚、若大股較己丙比大句甲辰。

Using [the] small major-cathetus chariot-beam HI [to] compare unto [the] small minor-cathetus FG, [is] as [the] great major-cathetus chariot-beam FC compared unto [the] great minor-cathetus AO.

In modern notation,
\begin{aligned} \frac{\colb{HI}}{\colr{FG}} &= \frac{\colg{FC}}{\colv{AO}} \\[\tallspace] \colv{AO} &= \frac{\colr{FG} \cdot \colg{FC}}{\colb{HI}}. \end{aligned}

既得甲辰、加表高辰乙（即丙丁）、得甲乙、爲島高。

Having gotten AO, add [the] post height OB (which is CD), [to] get AB, [which] be [the] height of [the] island.

In modern notation,
\begin{aligned} AB &= \colv{AO} + \colr{OB} \\[\tallspace] &= \frac{\colr{FG} \cdot \colg{FC}}{\colb{HI}} + \colr{OB} \\[\tallspace] \textq{island height} &= \frac{ \textq{\colr{post height}} \textq{\colg{post separation}} }{ \textq{\colb{mutual excess}} } + \textq{\colr{post height}}. \end{aligned}
Remember
\begin{aligned} \textq{\colb{mutual excess}} &= \colb{HI} \\ &= HG - IG \\ &= HG - ED \\ &= \textq{rear post backward walk} - \textq{front post backward walk}. \end{aligned}

小股較辛壬比分股壬庚、若大股較己丙比分股丙辰（即丁乙）、得島去前表之遠。

[The] small major-cathetus chariot-beam HI compared unto [the] split major-cathetus IG, [is] as [the] great major-cathetus chariot-beam FC compared unto [the] split major-cathetus CO (which is DB), resulting in [the] distance [that the] island [is] separated from [the] front post.

In modern notation,
\begin{aligned} \frac{\colb{HI}}{\colr{IG}} &= \frac{\colg{FC}}{\colv{CO}} \\[\tallspace] &= \frac{\colg{FC}}{\colv{DB}} \\[\tallspace] \colv{DB} &= \frac{ \colr{IG} \cdot \colg{FC} }{ \colb{HI} } \\[\tallspace] \textq{\colv{island--post separation}} &= \frac{ \textq{\colr{front post backward walk}} \textq{\colg{post separation}} }{ \textq{\colb{mutual excess}} }. \end{aligned}

《海島算經》 "[The] Sea Island Computational Classic"

Translation

Problem

Derivation

See also

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