The Grand
Astrologer’s platform and ramp: Forthcoming
in Third revision, 27 July 2012 Tina
Su-lyn Lim
林淑鈴 Donald B. Wagner 華道安
We are grateful to Karine Chemla for drawing our attention to Wang Xiaotong’s interesting text, to Jesper Lützen and Ivan Tafteberg for detailed comments on an earlier version, and to Jia-ming Ying 英家銘 and three anonymous reviewers for comments on earlier drafts of this version. ## AbstractWang
Xiaotong’s 王孝通的《緝古算術》主要在處理立體與平面幾何中使用到三次方程式的問題，而這
些方程式是用中國版的霍納法求出數值解。立體幾何的問題是給定立體體積與外部尺寸的某些限制, 然後再求解外部尺寸;
本文翻譯且分析四個問題. 其中三題是用分割法求解, 另一題則是用關於計算的推理求解, 並且很少觸及幾何上的考量.
王孝通著作中的問題本質上不能看成實用問題, 但這些問題介紹的數學方法對於官吏組織勞動力進行公共工程很有幫助.
(英家銘譯） Subject classifications: 01-02, 01A25. ## KeywordsWang Xiaotong, ## Introduction
Wang Xiaotong (late
6th–7th century AD) served the Sui and Tang dynasties
in posts concerned with calendrical calculations, and
presented his book, Printed in smaller characters in the text
are comments which generally give explanations of the
algorithms of the main text. The earliest extant
edition states that the comments are by Wang Xiaotong
himself, but we have noticed some differences in
terminology between the comments and the main text,[4] and feel therefore that the question
of the authorship of the comments should be left open.
Perhaps most but not all of the comments are by him. ## The textWe have treated the history of Wang
Xiaotong’s text in detail elsewhere.[5]
The oldest extant edition is from 1684,[6] and the best traditional critical edition is that
of Li Huang (1832).[7] The standard modern edition has long been
Qian Baocong’s [1963, 2: 487–527; important
corrections, 1966], but that of Guo Shuchun and Liu
Dun [1998] has much to recommend it. In the parts of
the text treated in the present article there is no
important difference between the two, and we generally
follow Qian Baocong’s version. The 1684 edition is marred by numerous
obvious scribal errors which must be corrected by
reference to the mathematical context. Guo and Liu
[1998, 1: 22] note that Li Huang introduced ca. 700
emendations to the text, and that Qian Baocong
followed most of these but introduced 20 new
emendations. The comments, written in smaller
characters than the main text and also often more
difficult in their mathematical content, are even more
subject to scribal errors. • The ## Solving cubic equations by ‘Horner’s method’In the problem solutions given by
Wang Xiaotong the coefficients of a cubic equation are
calculated, after which the reader is instructed to
‘extract the cube root’. Wang Xiaotong gives no
indication of how this was done, but it was obviously a
well-known procedure, for algorithms are described in
detail in the So much has been written about Chinese methods for extracting the roots of polynomials that it is not necessary to go into detail here. The algorithm is equivalent to ‘Horner’s method’ as sometimes taught in modern schools and colleges.[10] The first digit of the root is determined, the roots of the function are reduced by the value of that digit, the next digit is determined, and these steps are repeated until the desired precision is reached. The operation is carried out on paper in the Western version, in the Chinese version with ‘calculating rods’ laid out on a table. ## Problem 2: The Grand
Astrologer’s platform and ramp
The text of the problem and solution are translated at the end of this article.[11] Here we provide an overview using modern terminology. A tamped-earth platform and associated ramp are described, to be built by workers from two counties, A and B (translation, Sections 2.1.1 and 2.1.2. Two possible interpretations of the description are shown in Figure 1. Other interpretations are also possible, but the mathematics is the same whichever interpretation is followed. It quickly becomes
clear that the problem is not intended to reflect real
life: for example, the tamped-earth ramp is broader at
the top than at the bottom, and building such a
construction would be a heroic engineering feat.
Rather, a general mathematical method is introduced,
in an entirely unrealistic context, which will be used
many times in the rest of the book. The method is
analyzed in greater detail further below, in the ‘Concluding remarks’.
The
basic geometric situation is shown in Figure
2. The following quantities are given:[12]
Labour data is given which allows
the straightforward calculation of the volumes of the
two structures,
The dimensions of the two structures and of the two counties’ contributions are required. ## The dimensions of the platformThe first problem is to calculate Wang Xiaotong’s calculation proceeds as follows (translation, Section 2.3.1). Area for the Corner Truncated Volume for the
Corner Area for the Corner Heads, Truncated Volume for the Corner Heads, Determined Number, These names refer to one of the
basic solids in classical Chinese geometry, the
The cubic equation whose root is to be extracted is then
This equation has one real root,
Once In Figures 1–2 we have shown the
platform as a frustum of a right pyramid, but the
calculation is correct for a frustum of any
rectangular pyramid, and the names given to
intermediate quantities in the calculation suggest
that Wang Xiaotong may have derived (2) using a volume
dissection similar to that shown in Figure
4, in which one edge of the frustum is
perpendicular to the base.
The platform is
divided into pieces whose volumes are sums of products
of known quantities and powers of the unknown,
The ‘corner
= And no. IV has volume,
Summing,
= which is equivalent to (2). ## The contributions of the two counties to the platformThe next problem is to calculate Height for the Upper Width,
Height for the Upper Length,
These ‘heights’ are not equal to any dimensions in the geometric situation; they are simply intermediate results in the calculation. The height
This equation has one real root,
The remaining dimensions are then calculated by considering similar triangles: = 85 = 130 A comment
(translation, Section
2.3.3) gives a remarkable explanation of (6).
The text of the comment is corrupt as it has
come down to us. Luo Tengfeng (1770–1841) [1993, 1:
16a–18b]
has proposed an interpretation and a large number of
emendations to the text; we feel no real doubt that
his analysis of the underlying mathematics is correct,
even if his precise emendations may be less certainly
correct. In the translation we follow all of his
emendations. The comment uses some
geometric designations for intermediate quantities,
but the reasoning concerns calculations rather than
geometry, and thus might be considered almost purely
algebraic.
The
platform and the contributions of the two counties
(see Figure
5) are what is known in classical Chinese
mathematics as
The formula is not stated in the
comment but clearly is used. Substituting this
expression for = Doubling this gives =
The comment first
states that multiplying the Height for the Upper Width
by the Height for the Upper Length (
The comment introduces two more named quantities, Height for the Lower Length,
(10) Height for the Lower Width,
(11) These quantities correspond to no dimensions in the geometric situation, and at this point their numeric values are unknown; they are abstract quantities which figure in the algebraic reasoning which follows. They are not actually defined in the commentary; presumably the reader is expected to deduce their definitions by analogy with the Heights for the Upper Length and Width (see equations (4) and (5) above). The comment
states, ‘multiplying the Height for the Lower Length
by the Height for the Upper Width gives one of the
middle areas’. The ‘middle areas’ are This is followed by a statement that
This is true because, considering similar triangles, so that
Then ‘the middle area is composed of one small area and also an area obtained by multiplying the Height for the Upper Width by the truncated height’:
Further,
‘multiplying [. . .] by the Height for the
Upper Length gives one of the middle areas’. The other
multiplicand is missing in the text, but the context
indicates that it can only be the Height for the Lower
Width, so that the ‘middle area’ is
Finally,
‘multiplying the Height for the Lower Width by the
Height for the Lower Length gives two of the large
area’. The ‘large area’ is
Or,
In conclusion, ‘Thus there are two of
the area formed by multiplying the truncated height by
itself [ 2 _{7}
3 K_{6}h_{B}
3 K_{7}h_{B}
‘These, multiplied by the truncated
height [
## The dimensions of the rampThe next problem is to calculate Area for the Corner
Volume for the
Area for the Vertical and
Horizontal Edge
(
This equation has one real root,
The remaining dimensions of the ramp
are then easily calculated.
The names given to
intermediate quantities suggest that (13) was derived
using a dissection similar to that shown in Figure
6. The volume is divided into parts whose
volumes are sums of products of known quantities and
powers of
= = The ‘
= = The ‘corner
Combining gives the volume of the ramp:
6 = And from this (13) can easily be derived. ## The contributions of the two counties to the rampThe final problem is to calculate
This equation has one positive root,
The remaining dimensions are then easily calculated:
In this case Wang Xiaotong gives us
no clue as to how he derived his calculation, but the
dissection shown in Figure
7 will serve. A necessary implicit assumption is
that
Considering similar triangles, whence, and multiplying by ## Concluding remarksThe first and third problems
discussed here are dressed up as practical problems, but
clearly are not: No practical problem in the
construction of earthworks starts with a volume and the
## Volume dissectionsThe first and third of the problems analyzed here provide clues which allow us to guess at the volume dissections used by Wang Xiaotong in deriving his results (Figures 4 and 6). The fourth provides no clues, but a dissection along the same lines seems clear enough (Figure 7). How were the dissections communicated? There is no indication that Wang Xiaotong’s book contained diagrams of any sort, nor are diagrams of three-dimensional geometric situations known in other early Chinese mathematical works.[15] If we make the reasonable assumption that the book was intended as an adjunct to a teacher’s instruction, then an interesting question is how a teacher communicated these often rather complex geometric situations to a student. He may perhaps have possessed scale models of each problem, but study of an earlier text suggests a simpler and more flexible means. The commentary by
Liu Hui 劉徽 (3rd century
AD) on the
each with
dimensions 1 × 1 × 1
If Wang Xiaotong,
or a teacher using his book, possessed a set of such
blocks, he could easily have demonstrated the
dissection of Figure
4, for example, by
building the structure shown in Figure
9. It would only be necessary for the student to
abstract from the difference in dimensions between the
actual geometric situation and the constructed model.
Each of the volume dissections described here divides an object into parts whose volumes are sums of products of known quantities and powers of the dimension to be calculated. Each involves some algebraic reasoning: for example, in the first problem, equation (3), = where (
and this is easily demonstrated by a
dissection of the area of a rectangle with dimensions ## More advanced reasoningThe calculation in the second problem seems not to lend itself to a volume-dissection derivation along the same lines as the other three, and a comment gives instead a derivation which depends entirely on reasoning about calculations. This is not easy. The commentator’s greatest problem is perhaps in the naming of abstract quantities. His usual way of naming an intermediate quantity in a calculation is to describe an actual calculation with known quantities; for example, he expresses (4) as, Multiply
the height of the platform by the upper width and
divide by the difference between the widths to make
the Height for the Upper Width. But he seems unable or unwilling to
give a name to a calculation with unknown quantities,
for example expressing (10) as ‘let Height for the Lower
Length denote the product of . . . divided by
. . .’. He does not explicitly define the
Heights for the Lower Width and Height ((10) and (11)
above), but requires the reader to deduce the
definitions of these terms by analogy with the Heights
for the Upper Width and Height ((4) and (5) above).
Similarly, he refers to concrete areas (in the plane In spite of the difficulties which the author of the comment confronts, he does succeed in outlining the considerations involved, so that a student would be able to obtain an intuitive understanding of the underlying reasoning. ## Problem 2: Chinese text and translationIn the Chinese text we indicate text
variants with the notation { Our mathematical comments are given indented in the translation, while a few philological comments are given in footnotes. ## [2.1. Problem]## [2.1.1.
The observatory platform]
假令太史造仰觀臺，上廣袤少，下廣袤多。上
下廣差二丈，上下
袤差四丈，上廣袤差三丈，高多上廣一十一丈。甲縣差一千四百一十八人，乙縣差三千二百二十二人，夏程人功常積七十五尺，限五日役臺
畢。 In the following see Figures 1 and 2. Suppose that the Grand Astrologer
builds a platform for observing the heavens. The upper
width and length are smaller, while the lower width and
length are larger. The
difference between the upper and lower widths is [ 1 County A sends 1418 corvée labourers
and county B sends 3222 corvée labourers. The regulation
for one man’s labour in summer is a constant volume of
75 [cubic] ## [2.1.2. The ramp]
羨道從臺南面起，上廣多下廣一丈二尺，少袤 一百四尺，高多袤 四丈。甲縣一十三鄉，乙縣四十三鄉，每鄉別均賦常積六千三百尺，限一日役羨道畢。二縣差到人共造仰觀臺，二縣鄉人共造羨道，皆從先給 甲縣，以次與乙縣。臺自下基給高，道自初登給袤。問臺道廣、高、袤及縣別給高、廣、袤各幾何？ A ramp rises from the southern
surface of the platform. The upper width is [ County
A [sends labourers from] 13 districts and county B
[sends labourers from] 43 districts. Each district is
uniformly assessed with a constant volume of 6300
[cubic] ## [2.2.] Answer
答曰： 臺高一十八丈，上廣七丈，下廣九丈，上袤一 十丈，下袤一十四 丈。 甲縣給高四丈五尺，上廣八丈五尺，下廣九 丈，上袤一十三丈， 下袤一十四丈。 乙縣給高一十三丈五尺，上廣七丈，下廣八丈 五尺，上袤一十 丈，下袤一十三丈。 羨道高一十八丈，上廣三丈六尺，下廣二丈四 尺，袤一十四丈。 甲縣鄉人給高九丈，上廣三丈，下廣二丈四 尺，{上 ／}袤 七丈{， 下袤一十四丈／}。 乙縣鄉人給高九丈，上廣三丈六尺，下廣三 丈，{下 ／}袤 七丈。 The platform: The contribution of
county A: The contribution of
county B: The ramp: The
contribution
of the men in the district of county A: The
contribution
of the men in the district of county B: ## [2.3.] Method
## [2.3.1. The dimensions of the platform]術曰：以程功尺數乘二縣人，又以限日乘之， 為臺積。又以上下 袤差乘上下廣差，三而一為隅陽冪。以乘截高，為隅陽截積{冪／}。又半上下廣差，乘斬上袤為隅 頭冪，以乘截高為隅頭截積。{所得／}并二積，以減臺積，余為實。以上下廣差并上下袤 差，半之為正數。加截上袤，以乘截高，所得，增隅陽冪加隅頭冪，為方法。又并截高及截上袤與正數，為廉法，從。開立方除之，即得上 廣。各加差，得臺下廣及上下袤、高。 Multiply
the regulation labour in [cubic]
Multiply
the difference between the upper and lower lengths [ Halve the difference between the
upper and lower widths [ Add the two volumes [ Add the difference
between the upper and lower widths [ Add the truncated upper length [ Further add together
the truncated height [ Extract the cube root to obtain the
the upper width [
Add to each of the differences to
obtain the lower width, the upper and
lower lengths, and the height of the platform.
## [2.3.2. The contributions of the two counties to the platform]求均給積尺受廣袤術曰：以程功尺數乘乙縣 人，又以限日乘之， 為乙積。三因之，又以高冪乘之，以上下廣差乘袤差而一，為實。又以臺高乘上廣，廣差而 一，為上廣之高。又以臺高乘上袤，袤差而一，為上袤之高。又以上廣之高乘上袤之高，三之，為方法。又并兩高，三之，二而一，為廉法， 從。開立方除之，即乙高。以減本高，餘，即甲高。此是從下給臺甲高。又以廣差乘{之／乙}高，{以／如／以[22]}本高而一，所得，加上廣，即甲 上廣。又以袤差乘 乙高，如本高而一，所得，加上袤，即甲上袤。其甲上廣、袤即乙下廣、袤。臺上廣、袤即乙上廣、袤。其後求廣、袤，有增損者，皆放此。 In the following see Figure 2. The method to calculate the volume
[of earth] supplied in [cubic]
Multiply it by three and multiply by
the area of [the square on] the height [
Multiply the height of the platform
[
Multiply the height of the platform
[
Multiply the Height for the Upper
Width [
Add the two heights
[
Extract the cube root to obtain [
Subtract this from the original
height [ = 45 Multiply the difference between the
widths [ = 85
Multiply the difference between the
lengths [ = 130
The upper width and length of [the
part built by] county A are the lower width and length of [the part built by]
county B [ If
later [one wants to] calculate the width and length,
[when] there are more or fewer [counties], in all
cases [the procedure] is the same. ## [2.3.3. Comment: Derivation of county B’s contribution to the platform]（此 應{三／六}因乙積，臺高再乘，上下廣差乘袤差而一。又 以臺高乘上廣，{／ 廣差而一，}為上 廣之高。又以臺高乘上袤，{／袤差而一，}為上袤之高。{／以上廣之高乘上袤之高／相乘}為小冪二。{因／因／又}下袤之高{／／乘上廣之高}，為中冪一。凡下袤、下廣之高即是截高與上袤、{與／／與[24]}上廣之高相連并數。然{此／此／則}有中冪定有小冪一，又有上 廣之高乘截高為冪{各／}一。又下廣之高乘下袤之高為大冪二。乘上袤之高為中冪一。其大冪之中{又／又有／有}小冪一，復有上廣、上袤之 高{為中冪／}各乘截高為{中／中／}冪各一。又截高自乘，為冪一。其中冪之內有 小冪一。又上袤之高乘截高為冪一。然則截高自相乘為冪二，小冪六。又上廣上袤之高各三，以乘截高為冪六。令皆半之，故以三乘小 冪。又上廣上袤之高各三，今但半之，各得一又二分之一，故三之二而一。諸冪{／乘}截{／高}為積尺。） This reflects multiplying the
volume [ 2 ×
Further, multiplying the height of
the platform [ These are the same calculations as above, equations (15) and (16). Multiplying these together gives
two of the small area [
Further, multiplying the Height for
the Lower Length [ The comment does not define the Heights for the Lower Length and Lower Width, but they turn out to be Further, multiplying the Height for
the Lower Length [ The Heights for the Lower Length
and the Lower Width [
Thus it has been determined that
the middle area [ Further, multiplying [twice] the
Height for the Lower Width [ Multiplying . . .[26]
[ Within the large area [ Within the middle area [ Thus there are two of the area
formed by multiplying the truncated height by itself [ 2 _{7} + 3K_{6}h_{B} + 3K_{7}h_{B}
These, multiplied by the truncated
height [
## [2.3.4. The dimensions of the ramp]求羨道廣袤高術曰：以均賦常積乘二縣五十六 鄉，又六因為積。 又以道上廣多下廣數加上廣少袤為下廣少袤。又以高多袤加下廣少袤為下廣少高。以乘下廣少袤為隅陽冪。又以下廣少上廣乘之，為鼈隅{／積}。以減積，餘，三而一，為實。 并下廣少袤與下廣 少高，以下廣少上廣乘之，為鼈從橫廉冪。三而一，加隅{／／陽[27]}冪，為方法。又以三除上廣多下 廣，以下廣少袤、 下廣少高加之，為廉法，從。開立方除之，即下廣。加廣差即上廣，加袤多上廣於上廣即袤，加{廣／高}多袤即道高。 The method for calculating the
width, length and height of the ramp is: Multiply the
constant volume of the uniform assessment by the two
counties’ 56 districts and multiply by six to make [6
6 In the following see Figure 2. Add the difference between the upper
width and the lower width of the ramp [
Add the difference between the
height and the length [
Multiply this by the difference
between the length and the lower width [
Multiply by the difference between
the upper width and lower width [
Subtract this from the ‘Volume’ [6
Add the difference between the
length and the lower width [
Divide by three and add the Area for
the Corner [
Divide the difference between the
upper width and the lower width [
Extract the cube root to obtain the
lower width [
Add this [
## [2.3.5. The contributions of the two counties to the ramp]求羨道均給積尺，甲縣受廣、袤，術曰：以均 賦常積乘甲縣一十 三鄉，又六因為積。以袤再乘之，以道上下廣差乘臺高為法而一，為實。又三因下廣，以袤乘之，如上下廣差而一，為都廉，從。開立方除 之，即甲袤。以廣差乘甲袤，本袤而一，以下廣加之，即甲上廣。又以臺高乘甲袤，本袤除之，即甲高。 The method to calculate the volume
[of earth] supplied in [cubic]
6 Multiply this twice by the length [
Multiply the lower width [
Extract the cube root to obtain [
Multiply the difference in widths [
Multiply the height of the platform
[ ## References
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Wang Xiaotong «Jigu suanjing» di er ti, di san ti
shuwen shuzheng 王孝通«缉古算经»第二题,第三题术问疏証 (Explication of
problems 2 and 3 in Wang Xiaotong’s Qian Baocong,
1998.
Suanjing shishu 算经十书, Li Yan Qian Baocong
kexueshi quanji 李俨钱宝琮科学史全集, vol. 4. Liaoning Jiaoyu
Chubanshe, Shenyang. Revised version of Qian Baocong
1963. Rees,
P. K., and Sparks, F. W., 1967. College algebra.
McGraw-Hill, New York. Shen
Kangshen 沈康身, 1964. Wang Xiaotong
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and commentary. Oxford University Press / Science
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http://books.google.dk/books?id=eitjhrgtg6yc. Volkov,
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Rongbin 王荣彬, 1990. Wang Xiaotong
«Jigu suanjing» zizhu yiwen jiaobu 王孝通«缉古算经»自注佚文校补 (Reconstruction of
the missing text in Wang Xiaotong’s commentary to his
[1] [2] ‘Nine
chapters of the mathematical art’ is a common
translation of this title. [3] Chemla and
Guo [2004: 387–457, 661–745]; see also Cullen
[1993]; Shen Kangshen et al. [1999: 250–306,
439–517]. [4] E.g. in the
commentary to problem 3, Qian Baocong [1963: 506]. [5] Lim and Wagner [forthcoming]; see also
e.g. Qian Baocong [1963: 487–491]; Guo Shucun and
Liu Dun [1998, 1: 21–22]; He Shaogeng [1989]. [6] This is the [7] On Li Huang’s
edition see Huang Juncai [2009]. [8] The only
exceptions to this statement appear to be the highly
speculative reconstructions of the fragmentary
commentary to Problem 17 by He Shaogeng [1989] and
Wang Rongbin [1990]. [9] Chemla &
Guo [2004: 322–335, 363–379]; Shen et al. [1999:
175–195, 204–226]; Wang & Needham [1955]; Lam
Lay Yong [1970; 1977: 195–196, 251–285; 1986];
Libbrecht [1973: 175–191]; Chemla [1994]; Martzloff
[1997: 221–249]. [10] See e.g. Rees
& Sparks [1967: 294–297] as well as numerous
pages on the World Wide Web. Horner [1819] presented
a procedure for approximating roots of any
infinitely differentiable function, but modern
descriptions of ‘Horner’s method’ consider only the
special case of polynomial functions. [11] Andrea Bréard
[1999: 95–99, 353–356] gives an abridged translation
of the text and an analysis of the first of the four
problems along the same lines as ours. Her analysis
uses a dissection into a larger number of pieces and
consequently a smaller amount of algebraic
reasoning. She seems to have been the first to note
that the names given by Wang Xiaotong to
intermediate quantities give clues to how the
calculations were derived. [12] [13] See e.g. Chemla
& Guo [2004: 390, 434, 436–439, 826–827, 912]. [14] And indeed,
Wang Xiaotong’s six problems on right triangles [Lim
& Wagner forthcoming] seem to be entirely
without practical applications. [15] A possible
exception is in Liu Hui’s explanation of the
cube-root algorithm in [16] Wagner
[1979]. [17] Lam Lay Yong
[1970]. [18] [19] In
keeping with the writing fashion of his time, in
which multisyllabic words were considered inelegant,
Wang Xiaotong abbreviates the terms [20] [21] Commentators are divided on the function of the word [22] Guo Shuchun
& Liu Dun [1998, 2: 4, n. 14]. [23] The point of this redundant sentence is not apparent. Perhaps it is a comment by another hand. [24] Guo Shuchun
& Liu Dun [1998, 2: 4, n. 19]. [25] [26] The
multiplicand is missing in the text here. The
mathematical context indicates that it is [27] Guo Shuchun
& Liu Dun 1998, 2:4, n. 26. [28] The height of the ramp, dulian 都廉 is occasionally
used by Wang Xiaotong for the quadratic coefficient
instead of the more usual lianfa 廉法. We have been unable to determine whether the
two terms are exact synonyms, or differ in meaning in
some way. |