ABSTRACT
Zu Chong-Zhi ‘c‰«”V's main writing, ZhuiShu<<’Ôp>> was lost in the Song pe
riod, so we cannot know his method of computing ƒÎ. There is now, however, Da M
ing Li<<‘å–¾—ï>>(Da Ming Almanac). Its new device is the method of computing a
tropical year, which uses the concept of `lu' —¦(linear ratio), a Chinese tradit
ional one.
So the present writer set up by a hypothesis that the method of Zu Chong-Zhi
's computing ƒÎ and Liu Hui's one was the same. If Zu Chong-Zhi used same meth
od and computed down to seven decimal places, he had to compute 12288 polygonal
circumference lengths. But even if the figures were increased, this method cann
ot compute down here because it used fixed point operation, not floating point o
peration. So he perhaps used the formula of Liu Hui, and had the approximate va
lue ofƒÎ.
KEY WORDS
computing a tropical year, the concept of `lu' —¦(linear ratio), fixed point o
peration
INTRODUCTION
It is said that Zu Chong-Zhi was a prominent scientist in the pages of histo
ry of Chinese traditional science. However, because of the loss of his main wri
ting Zhui Shu <<’Ôp>>(*1), it is impossible to grasp his mathematical achieve
ment directly. But from the various fragments of historical materials, we are a
ble to assume that his work reached a high level. Li Chun-Feng —›~•—, the ma
thematician in the Tang “‚ period, estimated Zu Chong-Zhi highly in particular
and mentioned in Sui Shu <<ä@‘>> (History of Sui Dynasty) Lu Li Zhi <<—¥—ïŽu>>
(Annals of Musical Rules and Calendars) as follows.
ŒÃ”V‹ãÉAš¢Žü—¦ŽOAš¢œl—¦ˆêA‘´p‘`–‘AŽ©—« A’£tA—«‹JA‰¤”×A”牄@
”V“kAŠeÝV—¦A–¢äj܈£A‘v––A“ì™BœnŽ–Žj‘c‰«”VAXŠJ–§–@AˆÈš¢œlˆê‰à¨ˆê
äAš¢Žü‰mÉŽOäˆêŽÚŽl¡ˆê•ªŒÜ—ЋãŸ|“ñ•bŽµšB ÉŽOäˆêŽÚŽl¡ˆê•ªŒÜ—ЋãŸ|“ñ
•b˜ZšA³É݉m “ñŒÀ”VŠÔB–§—¥Aš¢œlˆê•Sˆê\ŽOAš¢ŽüŽO•SŒÜ\ŒÜB–ñ—¦Aš¢œl
ŽµAŽü“ñ\“ñB–”ÝŠJ·™pAŠJ·—§AŒ“ˆÈ³•‰(*2)ŽQ”VBŽw—v¸–§AŽZŽ”VÅŽÒ–çB
Š’˜”V‘A–¼à¨<<’Ôp>>A›{Š¯”œ”\‹†‘´[‰œA¥ŒÌ”pŽ§•s—B
(historical material 1)
According to this material, it seems that he excelled in study of the quadra
tic and cubic equations, that is, the circles and the spheres as `kai-cha-mi' ŠJ
·™p and `kai-cha-li' ŠJ·—§(*3). Especially, the circular constant which was co
mputed as:
3.1415926 ƒƒÎƒ3.1415927 \\\\\\(0.1)
This paper is a hypothesis about the method of computing the circular consta
nt. We would like to consider if he followed Liu Hui—«‹J's method or thought up
his own new method.
CHAPTER ‚P
The Method of Computing The Winter Solstice in Da Ming Li
The historical materials about Zu Chong-Zhi are extremely limited. For exa
mple, it is said that Zu Chong-Zhi wrote the later half of Jiu Zhang Suan Shu <<
‹ãÍŽZp>>(Nine Chapters in Mathematical Art) , chapter 1, question 32, with not
e by Liu Hui, namely the past after `Jin Wu-ku' ç•ŒÉ (weapon-wearhouse of the
Jin dynasty)(*4), but there are the counterarguments, so we cannot conclude that
it was Zu Chong-Zhi's note.
At first, we consider Da Ming Li which is said as reliable material of Zu C
hong-Zhi's work. There are many historical materials about Zu Chong-Zhi, howeve
r Da Ming Li is almost the only one in which we can confirm as Zu Chong-Zhi's wo
rk, because he is often mixed up with Zu Geng ‘c , his son.
Therefore, considering the mathematical part of Da Ming Li , especially how
to compute the point of the winter solstice, we will search for Zu Chong-Zhi's
revolutionary achievement such as `There are no scholar understanding its essenc
e.' (historical material 1).
Da Ming Li is the epoch-making almanac. The gist is two improvements. One
is taking into consideration the precession of the equinoxes, it is first one in
China.
The other is changing the cycle of leap years, from 9 times per 17 years to
4 times per 391 years. The most important improvement is this one. He needs
to discover the precession of the equinoxes for computing exact one tropical yea
r. He computed a tropical year to create a new cycle. So he had to fix the sta
rting point of a tropical year, it is the point of winter solstice.
Because it was criticized furiously by the conservative scholar such as Tai
-Xing‘Õ–@‹»(*5), Zu Chong-Zhi described the method of computing the point of
e winter solstice.
In ancient China, the seasons were known by measuring the shadow-length of t
he sun using `biao' •\ (gnomon)(*6). When the shadow-length is the longest, it
is the winter solstice, and when it is the shortest, it is the summer solstice.
And since the tropical year starts from the winter solstice, maybe it is easy
to measure than the summer solstice. But the point of the winter solstice is no
t always a meridian passage, therefore even if the day of the winter solstice co
uld be fixed, the point of it could not be set.
Zu Chong-Zhi measured the shadow-length three times, before and after the wi
nter solstice and computed the time of the point,
‹’‘å–¾ŒÜ”N\ŒŽ\“úA‰eˆê䎵¡Žµ•ª”¼A\ˆêŒŽ“ñ\ŒÜ“úAˆê䔪¡ˆê•ª‘¾A“ñ
\˜Z“úAˆê䎵¡ŒÜ•ª‹AÜŽæ‘´’†A‘¥’†“V“~ŽŠA‰žÝ\ˆêŒŽŽO“úB‹‘´”a”ÓA—ߌã
“ñ“ú‰e‘ŠŒ¸A‘¥ˆê“ú·—¦–çB”{”Vਖ@A‘O“ñ“úŒ¸AˆÈ•S˜©”Vਛ‰A“¾“~ŽŠ‰ÁŽžÝ–é
”¼ŒãŽO\ˆê(*7)B (historical material 2)
namely it is as follows(*8).
DATE (‚˜) THE SHADOW-LENGTH (`chi' ŽÚ) (y )
10th October 461 (x1) 10.775 (y1)
25th November 461 (x2) 10.8175 (y2)
26th November 461 (x3) 10.75083 (y3)
TABLE-1 TIME AND THE SHADOW-LENGTH
And Zu Chong-Zhi drew the conclusion s as follows;
`yi ri cha lu'ˆê“ú·—¦ (ƒ¢y)
(The ratio of the mutation of the shadow-length in a day)
‚™‚Q | ‚™‚R
10.8175 |10.750830.0667
Then he assumed that the shadow-length would mutate by this rate from 10th,
Oct. to 26th, Nov.(*9).
The difference between the point of winter solstice
and the middle point of ‚˜‚P and ‚˜‚Q (3rd, Nov. A.m.0)
¬ii‚™‚Q|‚™‚Pj^ƒ¢‚™j \\\\\\\\\\\\(1.1)
¬0.31873 (day)
Then, because a day is 100 `ke' (ˆ0.24 hours), he omitted the decimal
place and got 31 `ke'.
So as stated above, Zu Chong-Zhi regarded time and the shadow-length as the
linear function. But the real shadow-length is exprssed by the following funct
ion.
‚™‚ˆE tankƒÓ|asin(sinƒÃEsin ‚˜')l \\\\\(1.2)
‚ˆ‚W (`chi')
x'x/365.242199~360 (deg.)
Then, the (1.1)-formula is only the approximate formula.
Like this, in the mathematics of ancient China, it is the traditional method
to compute the approximate value of complicated function with `lu' —¦ (linear r
atio).
For example, in Jiu Zhang Suan Shu, chapter 7, question 11, the exponential
function was regarded as the linear one, and about the approximate value of the
follow function,
¬˜g‚O‚P¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
¬ i‚P|‚QX ji‚U|‚Q Xj ¬
¬ ‚™ \ \\\\\(1.3) ¬
¬ ‚QX ¬
¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
it was solved by, the approximate one,
‚™‚R¬‚˜|‚W \\\\\\\\\\\\(1.4)
This is the typical case(*10).
As the many scholars point out, `lu' is the foundation of traditional mathem
atics in China from the Han Š¿ dynasty(*11), and the method of computing the poi
nt of the winter solstice by Zu Chong-Zhi is the practical one based on this `lu
', that is, the linear relation.
CHAPTER ‚Q
Computing The Circular Constant
Da Ming Li is novel as the almanac and the first revolutionary one of ancien
t astronomy in China that was completed in the HanŠ¿ period. One of the most im
portant revolutions was changing the cycle of leap years. When `7 times per 19
years' up to that time was corrected to `144 times per 391 years' it was done by
computing the correct tropical year. In short, Da Ming Li was based on that me
thod to compute the tropical year, which was stated in chapter 1. As stated abo
ve, such computing-method was the traditional one and it is possible say there w
ere no revolutionary achievements about in the mathematics.
Therefore, it is possible that computing ƒÎ was also based on the traditio
nal method. Not only being possessed of the historical material 1, it is necess
ary to consider the traditional method, that is, Liu Hui's. Then, at first, we
will examine Liu Hui 's method.
(1) Liu Hui 's first method
At first, Liu Hui drew the area of the regular polygons that inscribed the c
ircle. It is evident that the area of the circle is more than that value. Next
, having drawn the area of its double angled polygon, he doubled the difference
between the original regular and this area, and added its twice areas to the reg
ular one. Then it is turned and that this area was longer than that of the circ
le.(See fig.1) Namely;
‚r2nƒ‚rƒ‚rn {‚Q~i‚r2n|‚rn j\\\\\\\2.1.1
or
‚r2nƒ‚rƒ‚r2n|i‚r2n|‚rn) \\\\\\\2.1.2
And Liu Hui computed the area from hexa-polygon to 126-polygon, further he d
etermined the correct 314 as the area of the circle.
‚r192 ƒ‚rƒ‚r96{‚Q~i‚r2n|‚rn j
314.64/625 ƒ‚rƒ 314.169/625 \\\\\\2.2
In other words, he decided the circular constant 3.14.
This method theoretically may well be able to be extended infinitely, but, i
n the process of Liu Hui's computing method, because the fractions were cut off
when the value extracted, the errors arose. Therefore it was impossible to comp
ute the correct value. Computing actually, when the polygon was 384-one (that i
s to compute one side of 92-polygon), the result is as follows.
314.88/625 ƒ‚r384 ƒ314.100/625 \\\\\\2.3
314.4/25
3927/1250
However this conclusion is the nearest to the real value, from that time on,
the errors increased. So, Liu Hui described as follow;
ác‹ˆêçŒÜ•SŽO\˜ZŒÊ”Vˆê–ÊA“¾ŽOç‹ã•S“ñ\Žµ”V™pAŽ§Ù‘´”÷•ªAÉ–’‹X‘RA
d‘´é„Ž¨B(*12) (historical material 3)
But it is opportune view that he continued to compute actually to the edge l
ength of 192-polygon.
(2) Liu Hui 's second method(*13)
Therefore, Liu Hui discovered the difference of the areas very much resemble
d the geometrical progression, and drew up the following formula.
‚r‚r192 {ƒ¢‚r96~lim (ƒ° (ƒ¢‚r92^ƒ¢‚r49))n
314. 64/625 {105/625 ~lim ƒ° (¬) n
314. 64/625 { 35/625
¬314. 64/625 { 36/625
314.100/625 314.4/25iƒÎ3.1416j \\\\\\2.4
This is correct to three place of decimals.
(3) Zu Chong-Zhi's method of computing ƒÎ
However Zu Chong-Zhi drew the correct values to seven place of decimals, he
did not use quite the same method as Liu Hui's first method. So this table 2, u
sing seven figures of the significant digit, 786-polygon is the limit computed.
Its simple solution is to increase the significant figures. The diameter of th
e circle is regarded as 1 `zhang' ä (¬3 Ms.) following Sui-Shu Lu-Li-Zhi, the
n we can drew ƒÎ to eight places of decimals(*14). But the error increases and
the limit is to 6144-polygon. The value of that limit is as follows:
3.1415926 ƒƒÎƒ3.1415930 \\\\\\\\2.5
In other words, he could draw the lower limit facility by only computing the
outside circuit of the inscribed regular polygon.
The problem exists in the case of computing the upper limit. The (2.5)-formu
la is the conclusion of applying Liu Hui's first method. It differs from the v
alue in Sui-Shu Lu-Li-Zhi. But in the next case of 12288-polygon, the value is
smaller than the described 6144-polygon. Therefore, the following formula is di
scovered facility.
Then,
ƒÎƒ3.1415927
so,
3.1415926 ƒƒÎƒ3.1415927 \\\\\\\2.6
is drawn, and it is at one with that of Sui-Shu Lu-Li-Zhi.
And, using Liu Hui 's second method, if we add 1/3 ~ƒ¢‚bn which is the dif
ference (ƒ¢‚bn) of the finite progression to the original (‚bn), then the value
is drawn as shown in table 5,
ƒÎ3.1415927 \\\\\\\\2.7
Namely, it is upper limitation.
However, we should pay attention to that, there are no case that the common
difference (ƒ¢‚bn) is just 1/4 in table 5.
Therefore, it is impossible to draw the value of the infinite progression lo
gically. That is, it can he said only to substitute the value in the formula,
1/3 ~ƒ¢‚bn \\\\\\\\\\2.8
drawn by Liu Hui's second method.
Stated above, Zu Chong-Zhi's one was to compute the 6144-polygon for the low
er limitation, and to compute
a) ‚b1228ƒ‚b6144 then ‚bƒ31415927
or
b) Applying Liu Hui's second method.
for the upper limitation. And he drew 31415927.
Then, it proves that it is possible to compute the value drawn by Liu Hui wi
thout applying the revolutionary way.
You remember that Zu Chong-Zhi computed a tropical year by traditional metho
d, using `lu' as chapter 1. And we can see the (2.6)-formula is also the method
using `lu', because although they used the difference, they might not recognize
the non-linear function.
So, considering it was the traditional way to compute. It can be deduced th
at Zu Chong-Zhi's process to computeƒÎ might also be following the traditional o
ne.
CONCLUSION
Zu Chong-Zhi as the successor to Liu Hui
As the above mentioned, there was no revolutionary change between Liu Hui an
d Zu Chong-Zhi. Namely, they existed in the same paradigm. Zu Chong-Zhi magnif
ied and developed Liu Hui's method, but actually his method was the same basical
ly as that of Liu Hui.
Of course, as shown in table 4, a huge amount of work was needed to compute
ƒÎ to eight places of decimal, because it is necessary in drawing that to comput
e the four operations of arithmetic and the extraction of 14 figures. It is pos
sible to say that work was several, perhaps ten times as Liu Hui's work. But w
e should not evaluate the conclusion of this calculation but the work done by hi
m, that `yue-lu'–ñ—¦ (simple ratio) is 7/22 and `mi-lu' –§—¦ (minute ratio) is 3
55/113, as a mathematical achievement(*15).
But, that evaluation conflicts with Li Chun-Feng's description of evaluation
as historical material 1. Because it was thought that how to computeƒÎ was too
difficult for anyone to understand.
Then if we set historical material 1 from `yuan'š¢ (circle) to `fu' •‰ (minu
s) based on Qian Bao-Zong èA›¬, it is regarded as the description of the area,
the volume, and the formula. The achievement of the value, `Zu Geng Yuan-Li'
‘c Œ´— (the principle of Zu Geng), the ratio of the area of a section equals t
he ratio of the volume, is a well known fact. Then, Zu Chong-Zhi's evaluation m
ight be concerned with this `Zu Geng Yuan-Li'.
It is a great achievement for Zu Chong-Zhi father and son, to have drawn th
e volume of a sphere which Liu Hui failed to compute. As he said
—~蛌`‘[ˆÓAœœŽ¸³—BŠ¸•sè‹^AˆÈ˜Ö”\Œ¾ŽÒ(*16). (historical material 4)
Especially, Zu Chong-Zhi father and son were the first and only one who consid
ered the ratio of area of the section and that of the volume(*17). As for this
achievement, it is worth being extolled.
ANNOTATION
(*1):
It was lost between the Tian-Sheng “V¹ period (1023-1032) and the Yuan-Fen
g Œ³æ² period (1078-1085).
Li Di —›çŒ,`Zhui-Shu de Shi-Zhuang Shi-Dai Wen-Ti'ƒ<<’Ôp>>“IŽ¸™BŽž‘ã–â‘è
„ (`The Discussion of The Lost Age of Zhui-Shu'),in Shu-Xue Tong-Xun <<É›{’Êu
>> (Mathematical Interrogation),11,p.33-34 (1958)
(*2):
In original text, it is ` ³š¢ŽQ”V' but ` š¢'(circle) is mistaken as `•‰'(mi
nus).
Qian Bao-Zong èA›¬, Zhong-Guo Shu-Xue-Shi <<’†š É›{Žj>> (The History of C
hinese Mathematics),p.87-88 (1964)
(*3):
Yoshio Mikami said
‚rABE‚Œ^‚Q{iƒÎ\‚Qj‚Œ2 ^‚Q
+(‚`) ‚Œi‚’\‚Q‚Œj+(‚a) ‚Œi‚’\‚Q‚Œ)(‚Œ\‚Œ2 )/‚’
and (A) is `kai-cha-mi' ŠJ·™p and (B) is `kai-cha-li'ŠJ·—§(See fig.2).
Yoshio Mikami, `Seki Kowa no Gyouseki to Keihan no Sanka narabini Shina no S
ampo tono Kankei oyobi Hikaku'ƒŠÖF˜a‚Ì‹ÆÑ‚Æ‹žâ‚ÌŽZ‰Æ•À‚Ñ‚ÉŽx“ß‚ÌŽZ–@‚Æ‚ÌŠÖŒW
‹y‚Ñ”äŠr6)„(`The Relation and Comparison of Kowa Seki's Work, The Mathematician
s in Kyoto and Osaka, and Chinese Mathematics') inToyo Gakuho <<“Œ—m›{•ñ>> (Jour
nal of the Orient), 22 (1935)
(*4):
Li Di —›çŒ,`Jiu-Zhang-Suan-Shu Zheng-Ming Wen-Ti de Gai-Shu' ƒ<<‹ãÍŽZp>>
‘ˆ––â‘è“IŠTq„ (`Compendium of the Dispute about Nine Chapters in Mathematical
Art') in Jiu-Zhang-Suan-Shu yu Liu-Hui <<‹ãÍŽZp—^—«‹J>> (Nine Chapters in Mat
hematical Art and Liu Hui),p.35 (1982)
(*5):
Tai Fa-Xing ‘Õ–@‹»(414-465): He was a `Tai-Zi Lu Ben Zhong-Lang-Jiang' ‘¾Žq
—·æÊ’†˜Y« (a general of the Crown Prince Guards) at that time.
Song Shu <<‘v‘>> (History of Liu-Song Dynasty), vol. 94.
(*6):
It was drawn by using `biao'•\(gnomon) which is 8 `chi' ŽÚ( ¬30 cm)(See fig
.3). The basis of the `Huang He' ‰©‰Í(Huang He River), which is the birth place
of the ancient civilization in China, is stated in 35 N.L.. and around this nei
ghborhood. There are two cases that the shadow length is 6 `chi' ŽÚi¬30 cm j
. In other words, it becomes the right angled triangle that has the simplest in
terval ratio, that is
shadow-length :height : hypotenuse
‚R : ‚S : ‚T
On the Yuan Œ³ period, the measuring apparatus which had a height of 40 `chi
' ŽÚi¬cm) was constant for the purpose of decreasing the relative error, and i
ts height set a multiple of 4. This is `guan xing tai' ŠÏ¯‘ä(astronomical obse
rvatory) in Deng-Feng pref., He-Nan prov., China’†‘‰Í“ìÈ“o••Œ§(See fig.4).
(*7):
Song Shu <<‘v‘>> (History of Liu-Song Dynasty), vol. 13, Lu Li Zhi <<—¥—ï
Žu>> (Annals of Musical Rules and Calendars).
(*8):
The values of table-1 are very correct. In trial, substituting these values
into the (1.2)-formula and drawing the values of ƒÓ, its result is as follows.
ƒÓ‚P31.6109475
ƒÓ‚Q31.71838269
ƒÓ‚R31.71285046
ƒÓav31.68072688
These values are nearly the same as Jian-Kang ŒšN, capital of Liu-Song —«‘v
(now Nan-Jing“ì‹ž 32.06 N.L.), or Nan-Xu-Zhou“ì™B, Zu Chong-Zhi's post (now Z
hen-Jiang ’Á] 32.28 N.L.)(See fig.5).
(*9):
This method is not Zu Chong-Zhi's original but based on `Si Fen Li' Žl•ª—ï(Q
uarter Almanac) in the HanŠ¿ period. It failed to measure the first day of Wint
er and Spring in the 3rd year of the Xi-Ping period à”•½‚R”N(A.D.174), owing to
the long interval of measuring.
(*10):
Bai Shang-Shu ”’®š, Jiu-Zhang-Suan-Shu Zhu-Shi<<‹ãÍŽZp’Žß>>(Commentar
y of Nine Chapters in Mathematical Art),p.243 (1983)
(*11):
Li Ji-Ming —›Œp–å `Jiu-Zhang-Suan-Shu Zhong de Bi-Lu Li-Lun' ƒ<<‹ãÍŽZp>>
’†“I”ä—¦—˜_„(`The Ratio Theory in Nine Chapters in Mathematical Art') in Jiu-Z
hang-Suan-Shu yu Liu-Hui<<‹ãÍŽZp—^—«‹J>> (Nine Chapters in Mathematical Art an
d Liu Hui),p.245 (1982)
Guo Shu-Chun Šs‘t, `Jiu-Zhang-Suan-Shu he Liu Hui Zhu zhong zhi Lu-Gai-Ni
an ji Ji Ying-Yong Shi-Xi'ƒ<<‹ãÍŽZp>>’†”V—¦ŠT”O‹y‘´‰ž—pŽŽÍ„(`Nine Chapters
in Mathematical Art and the Conception of the ratio of Circumference to Diameter
in Liu Hui's Annotations') in Ke-Xue-Shi Ji-Kan<<‰È›{ŽjWŠ§>>(Collected Papers
of History of Mathematics), 11,p.21 (1984)
(*12):
Jiu Zhang Suan Shu<<‹ãÍŽZp>>(Nine Chapters in Mathematical Art), chapter
1, question 32, Liu Hui—«‹J's note.
(*13):
Yoshio Mikami ŽOã‹`’j, `Seki-Kowa no Gyoseki to Keihan no Sanka narabini Sh
ina no Sanpo tono Kankei oyobi Hikaku 6)' ƒŠÖF˜a‚Ì‹ÆÑ‚Æ‹žâ‚ÌŽZ‰Æ•À‚Ñ‚ÉŽx“ß‚Ì
ŽZ–@‚Æ‚ÌŠÖŒW‹y‚Ñ”äŠr6)„ (`The Relation and Comparison of Kowa Seki's Work, Math
ematicians in Kyoto and Osaka, and Chinese Mathematics') in Toyo Gakuho <<“Œ—m›{
•ñ>> (Journal of the Orient), 22,(1935)
(*14):
If we compute to 24576-polygon using the (2.1)-formula, setting the signific
ant places as 9 figures. Then:
ƒÎ24576 ƒƒÎƒƒÎ12288 {‚Q~ iƒÎ24576 |ƒÎ12288 j
3.14159262ƒƒÎƒ3.1415925 {‚Q~i3.14159262|3.14159252j
3.14159262ƒƒÎƒ3.14159272
But it is the case that he set the significant place as 9 figures constantly
, then it is impossible to continue computing by increasing the error actually.
(*15):
ƒÎ expressed by the decimal is described to have drawn the value of the frac
tion by applying
1)`Diao Ri Fa' ’²“ú–@
2)`Lian Fen Shu Fa' ˜A•ªÉ–@
3)`Qiu Yi Shu' ‹ˆêp
by a lot of investigators. They are reliable.
Mai Rong-Zhao ”~‰hÆ, `Liu Hui yu Zu Chong-Zhi Fu-Zi' ƒ—«‹J—^‘c‰«”V•ƒŽq„ (
`Liu Hui versus Zu Chong-Zhi and His Son') in Ke-Xue-Shi Ji-Kan <<‰È›{ŽjWŠ§>>(C
ollected Papers of History of Mathematics), 11,p.21 (1984)
(*16):
Jiu Zhang Suan Shu<<‹ãÍŽZp>>(Nine Chapters in Mathematical Art) chapter 4
, question 24, Liu Hui—«‹J's note.
(*17):
In ancient China, it is evident that the relation between the area and the v
olume was not studied strictly because the units of length, area, and volume wer
e all the same.
¬˜g‚O‚Q¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
¬ c' ¬
¬ a a' ‚` l ‚a ¬
¬ b' ¬
¬ c b r ¬
¬ ‚’ ¬
¬ ¬
¬ ‚n ‚n ¬
¬ Fig.1 Liu Hui's First Method Fig.2 Mikami's interpretation ¬
¬ ¬
¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬¬
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