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o
INTRODUCTION by Guo Shuchun and Guo Jinhai
he Si Yuan u Jan ( Jade Mir or of the Fou Unknowns 1
1
j
was wit-
ten n h ee ee pars by Zhu Shije and publshed n 1303 duing he Yuan dynasy. It s the most advanc ed wok o tadtioal Chn ese mathematcs
c : ' ;
0
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n ode t uestand its status n th e histoy o mah ematcs in China and inteaionaly, w e ist need to noduce bely th e geneal stuation o tadiona Chnese mathematics be o e the ade Mrrr of the Four Unknowns was witt en Tadtiona Chine se math ematcs s one o he mos advanced basc dis cpes in ancie China and om the d c entu BCE to the b einning o the 4th centuy CE was th e eadng and most advanc ed in th e wod o mo e than 700 y eas Compa ed with the state o mathematcs n th e medi e va
he le adng and most W est the Jad e Mi o cetainy epes ens one o h
signicant mathematcal deveopm ents on the wold math ematica sene ath Although Chnese hst wtten n chaactes s vey eay no math eatca woks have suvived om the e o the eaest dynas dynast tes, ncuding the Xa Shang and W este hou no om the Spng and Autumn p erod as we he e o e, the natu e o the dev elopment o Chinese mathe matics dung these p eiods is not ceay und estood At p es ent we know that he Chnese Ch nese ceated he mos onenient ans ueati the wod: h e
41
• R .
decima pace-vaue system accompanied by what a the ime was the most mo st advanced cacuation cacuation method as we: we : counting-rods. These ae two of the
1
wo's most signicant accompishments The chaacte of taditional Chi nese mathematic mathematicss is diffeent fom ancient Geek mathematics, mathematics, but e num systm and mhod o ompuation ae two innovtions that have an in sepaabe connection The Waing States peiod and the Qin and Han dynasties estabished he foundations fo unifying China with espect to systems of oganization usiness pactices pactices terrtoia boundaes cutua cutua poducts inteecta attes et. Fho thy stbshd the ondtons fo hnese aon an aso fo taditiona Chines Chinesee mathematics The Suan Shu Shu ( Book of Athmetic )I 21, the Zhou Bi Suan Jing Aihmetca assic of the Gnomon and he Cicua Paths of eaven and the u hang Suan Shu The 3
Nine Chaptes o Mathematca Pocedues wee nshed beween the 4
Wng tes pei peiod od and beginning of he he Wese an dynasty any 42
aspects of Chinese mahematic wee ahead o the wod eading eves an some eceeded them fo moe than a thousand thousand years incuding computa tions wi he four basic athmetic opeations fo factions the aithmetic of popoion and poportiona distibution distibution the ue of excess and defciency the appoimation of squae and cue oots soutions of simutaneous tons es es o ddon nd sbon o os os nd nga ngate te ns inding socaed "Pythagoean pes etc ematical Proedures
Te Nne Chpters of Math-
estabshed the basic famewo fo taditiona Chinese
on cmputatons mahematics t cosey eates eoy and pactice ocusing on chaacte of its aitmetic is suctua and mechanica mechanica whch inuence the chaacte th dvopmnt of hinese and oena mathmtis mathmtis fo th net two tho
sand years or so thereaer. I aso shows tha alongside the sgnicant mah ematica deveopens in the Mediteanean, Chna ( and aer ndia and the slamic Near East ) were aso portant centres o matheai matheaical cal activy Ch nesee mathematcs also represens a diferent ephasis, oe epoying he nes relaonshps between quanies raer an eoetrc space and ogca deducon. reecs as we mahematics ui uing ng away ro an ephasis
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prary upon deducive ogc to an arhec endency coinng nduc ve and deducve ogc From te end o the Eastern Han dynasty to the We Jin and Souhe and Nohe dynases economy and polits ·were based upon a syste o aora sero sero The pracice amog amog schoars o deban deban the "three mysteries hese were the san xuan, which mainy included he Book of Change he Lo Zi and Zhuang Z repaced the trvalj ue ( cassca
sdies ) in in the wo Han perods. Chinese siety deveoped deveoped no a new phase. The accumuation o mahematica knowedge te pr o debae aon scholars, and he inuence o ois ough spued Lu u o the We Nne Chapers of Mahemacal perod to wre his Commenta on "The Nne Proedres. He analyzed e pncpes, and sed he dagrams o dispay
heir coponens, and gave any rgorous athematca deintons He used deductive ogc as hs man et to prove e e maeacal prncpes prncpes Nine Chae of Mahemaa Poe Poeu, u, e os opreensive o The Nine
Chnse aematics His proo o the orua or he wor o raditional Chnse aea o a circe and iu Hu' s heorem [ were the irs n China to intro 5
duce he concep o li and the ehod o ininiesias ino mahematca proos Hs metod o cutting e circle wiout imit and hs dea o solving solvi ng wi inniesimals qu we hu ) prided he oundaon o te aua-
43
tion of in Chinese mathematics. In fact Liu Hui' Hui' s theorem for detern ing the voumes of soids is eated to the third probem brought foward by 1862 62 - 19 1943 43), who presented his famous ist of 3 David iert ( 18 "unsoved proems for the the 0th 0th century at the the Inteationa Congress of Mathematicians hed in Paris in 00 iu ui s Hai Dao Suan Jing Manua ) l was devoted to Chinese techniques Sea Isand Mathematia Manua) for measuring heights and dstances for which the method of doube differ ences was the main method used one that as not supassed fo more than 3 years, unti the introduction of Weste methods for the measurement o heghts and dstances at the end of the Ming dynasty. Another andmak contbution to Chinese mathematcs was the Zhu Shu y Zu Chongzhi ( 4 5 CE) CE) a more profound profound book witten in the Southern dynasty It is a pity that that ocers at the Suan Suan Xue Guan ( Mathematics Bureau) Bureau) durng the Sui and Tang dynasties ere unabe to understand it and as a resut it was eventuay aoished Subsequenty h ook as ost ost t peset we ony kno to of is contrution advanced by Zu Chonghi and hs son Zu Gengzhi based on the foundations o iu ui s achievement One of these was the computation o the vaue of to eight decima paces as we as a more accurate ratio for i� ; the second was Zu Genghi s theorem equivaent to Cavaier s theorem in the West) y which Zu Gengzhi de termined the coect fomua or the voume of a sphere Other cassic texts of ancient Chnese mathematics gave soutions for epressions of conguence conguenc e of frst degree as did the Sun Zi Suan Jn ( Master Suns Mathematica andd the method of the undred Fows in Manua)) ( 4th centuy CE) an Manua the Zhan Qjan Suan Jn ( Zhang Qiujians Mathematica Cassic th enury C [ l, among others ,
ilj
-
6
7
8
9
44
:
10
j
Wth he grea developmen of agricuure, handicraf ndusry and nfuenced Chinese economy and polcs be comerce many new actors nfuenced ro d o he Tang dynasty. Durng he Song nd Yuan gnnng n he mdde rod dynases hese nfuences became more aure. Moreover he amosphere whn he crce o nellecua nellecua wa co1paravely ree. Ancen Chnese h cience and echnology echnology deveoped o new heghs In he rs half of he h cenury Ja Xan composed he Huang Di Jiu Zhan Suan Jin Xi Cao ( Deaied Expanaion of he Ancen Cassc Nne Chapers ) ( frs haf o h cenury CE ) n which he ade he soluon mehods of The Nine 12
hapte of Mathematical Procedures more abstrac He ceaed he mehod
C = -=
=
e : -= e=
:
ka fan zuo fa en yuan ( ogn o he ehod for exracing roo ha an's riange and he mehod of eng chen kai fan fa a mehod s Jia an's
.
for exracng roo usng addon an uplicaion slar s lar o Horner's mehod ) which represens he apex of Chnese mahemacs n he Song and Yuan dynases The l 3h cenury wa a prod durng whh the ot imporan h eacal works were handed down o he presen A ha me here were wo athemacal athema cal cenes on onee n n he souh souh anoher n he norh he lower reaches of the Yangze Rver were goveed by he Souhe Song dynasty where a cenre emerged whch whch ncluded ncluded Qn Juhao and and Yang Hu hey advanced oluons for equaons of hgher powers for lnea indeernate euaon nd impoved he ple ehod o cacuan r muplicaion and dvi 202 2 - 26 26 CE ) son among her an conbuons Qn Jushao ( a 20 copoed he Shu Shu Ju Zhang ahematcal Trease in Nne Secons ) ( 247 13 n whch he pu forward forward h mehod da an on on shu shu general da a, mehod thereb consummaed the ethod for expresons o con
45
gence of st degree and impoved he numecal souton soutonss of equations of hghe powes. Yang Hu composed he he Xiang lie Jiu Zhang Suan Fa ( Detaed Expanatons of the Agorhms n the Nne Chapte )( 126 ) 14 , and the Yang Hui Suan Fa ( ang Hu 's Mathemaca Mehods) 24725)[ 1� l among ohe works ang acheved consderabe success wth hs duo ji ji shu ( methods fo summng seres) and mpoved the smpe mehod
of cacuaon fo mutpcaton, mutpcaton, dvson dvson ec The nohe nohe mahemaca h u cene goveed by he Jn and uan dynases deveoped he ian yuan hu
( ceesa eement mehod fo sovng sovng equatons of hghe degee er an hu ( two eemens method and an yuan hu ( hee eemens meho meho gu u gu ng ng yn for sovng quadac and cubc equaons he kowedge of g (
measuemens of o f rgh anges and cces) and u ji hu ( mehods fo
Ha i Jing Jing ( Sea summng sees L e 92279 wote wo te he Ce Yuan Hai
Mro of Cce Cce Measuemens Measuemens 248 248! ! 16 and he Yi Gu Yan Duan ( Od Mathematcs n Expanded Secons Secons 5) 7 both of whch ee devoed to he method o ian uan shu ( ceesta eemen method) method ) Wang Xun 46
( 1235-1281) andGuoShoujng 231316 uizhacha
seres n he Shu Shi Caena. Zhu method for summaon of nte seres She, a a the end o he 3h century studed and acceped the best suts and methods fom the wo mahemaca centes n the souh and the north, and as a esut hs mathematca acheemens wee even geate than those of hs pedecessors
2
ZhuShe Zhu She aso now as angqng angq ng and Songng esided in an ( o
