1674:
55:
832:
The concept of negative numbers also appears in "Nine
Chapters of Arithmetic". In order to cooperate with the algorithm of equations, the rules of addition and subtraction of positive and negative numbers are given. The subtraction is "divide by the same name, benefit by different names. The addition
863:
Gou Gu mutual seeking discusses the algorithm of finding the length of a side of the right triangle while knowing the other two. Gou Gu integer is precisely the finding of some significant integer
Pythagorean numbers, including famously the triple 3,4,5. Gou Gu dual capacity discusses algorithms for
828:
does not discuss natural numbers, that is, positive integers and their operations, but they are widely used and written on the basis of natural numbers. Although it is not a book on fractions, the meaning, nature, and four operations of fractions are fully discussed. For example: combined division
887:
The "equations" discussed in the Fang Cheng chapter are equivalent to today's simultaneous linear equations. The solution method called "Fang Cheng Shi" is best known today as
Gaussian elimination. Among the eighteen problems listed in the Fang Cheng chapter, some are equivalent to simultaneous
883:
Completing the squaring and cubes can not only solve systems of two linear equations with two unknowns, but also general quadratic and cubic equations. It is the basis for solving higher-order equations in ancient China, and it also plays an important role in the development of mathematics.
864:
calculating the areas of the inscribed rectangles and other polygons in the circle, which also serves an algorithm to calculate the value of pi. Lastly, Gou Gu similars provide algorithms of calculating heights and lengths of buildings on the mathematical basis of similar right triangles.
859:
Regarding the direct application of the Gou Gu
Theorem, which is precisely the Chinese version of the Pythagorean Theorem, the book divides it into four main categories: Gou Gu mutual seeking, Gou Gu integer, Gou Gu dual capacity, Gou Gu similar.
833:
is "divide by different names, benefit from each other by the same name. Among them, "division" is subtraction, "benefit" is addition, and "no entry" means that there is no counter-party, but multiplication and division are not recorded.
376:
is an anonymous work, and its origins are not clear. Until recent years, there was no substantial evidence of related mathematical writing that might have preceded it, with the exception of mathematical work by those such as
237:(compiled throughout the Han until the late 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to
839:
gives a certain discussion on natural numbers, fractions, positive and negative numbers, and some special irrationality. Generally speaking, it has the prototype of the real number system used in modern mathematics.
888:
linear equations with two unknowns, some are equivalent to simultaneous linear equations with 3 unknowns, and the most complex example analyzes the solution to a system of linear equations with up to 5 unknowns.
248:
Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on by
856:
also discusses volumetric algorithms of linear and circular 3 dimensional solids. The arrangement of these volumetric algorithms ranges from simple to complex, forming a unique mathematical system.
934:
can be best concluded as "problem, formula, and computation". This process of solving applied mathematical problems can now be considered the standard approach in the field of applied mathematics.
221:
book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from
900:, or "9", means more than just a digit in ancient Chinese. In fact, since it is the largest digit, it often refers to something of a grand scale or a supreme authority. Further, the word
1673:
919:
stops short at the advancement of modern mathematics due to its focus on practical problems and inductive proof methods as opposed to the deductive, axiomatic tradition that Euclid's
829:(addition), subtraction (subtraction), multiplication (multiplication), warp division (division), division (comparison size), reduction (simplified fraction), and bisector (average).
409:
is an ancient
Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1983 when
1295:
Dauben, Joseph W. (September 1998). "Ancient
Chinese mathematics: the (Jiu Zhang Suan Shu) vs Euclid's Elements. Aspects of proof and the linguistic limits of knowledge".
1269:
Dauben, Joseph W. (1992), "The "Pythagorean theorem" and
Chinese Mathematics Liu Hui's Commentary on the 勾股 (Gou-Gu) Theorem in Chapter Nine of the Jiu Zhang Suan Shu",
1385:
Studies in the history of Indian mathematics: Papers from the seminar "Topics in the
History of Indian and Western Mathematics" held in Chennai, January–February 2008
2394:
852:
are mostly straight and circular figures because of its focus on the applications onto the agricultural fields. In addition, due to the needs of civil architecture,
294:
Most scholars believe that
Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time when
1823:
1852:
17:
1324:
Dauben, Joseph W. (2013). "九章箅术 "Jiu zhang suan shu" (Nine
Chapters on the Art of Mathematics) – An Appraisal of the Text, its Editions, and Translations".
904:, or "chapter", also has more connotations than simply being the "chapter". It may refer to a section, several parts of an article, or an entire treatise.
343:
step by step, in a manner which is clearly designed to give the reader confidence that they are reliable, although he is not concerned to provide
501:
took note of the book, but only with limited attention and only mentioning the usage of red and black rods for positive and negative numbers.
1751:
1512:
986:
A French translation with detailed scholarly addenda and a critical edition of the Chinese text of both the book and its commentary by
187:
2096:
1912:
880:
are very detailed. Through these discussions, one can understand the achievements of the development of ancient Chinese mathematics.
262:
280:
standard measures which are dated to 179 CE, but there is speculation that the same book existed beforehand under different titles.
1427:
Siu, Man-Keung (December 1993). "Proof and pedagogy in ancient China: Examples from Liu Hui's commentary on JIU ZHANG SUAN SHU".
1158:
298:
reached its final form. The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses
1682:
1641:
1587:
31:
1861:
1400:
1286:
999:
437:; and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. The
1616:
54:
2159:
1857:
1818:
1606:
2118:
2387:
1867:
1373:
981:
351:
manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians
2503:
2474:
1505:
447:
text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by
2518:
2013:
367:
records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century
2437:
1830:
1701:
876:
can be regarded one of the major content of ancient Chinese mathematics. The discussion of these algorithms in
69:
2089:
1791:
1722:
1646:
1553:
418:
91:
2508:
1591:
872:
The methods of completing the squares and cubes as well as solving simultaneous linear equations listed in
463:
1072:"A Radical Proposition on the Origins of the Received Mathematical Classic The Gnomon of Zhou (Zhoubi 周髀)"
972:
A full translation and study of the Nine Chapters and Liu Hui's commentary is available in Kangshen Shen,
429:
is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the
390:
194:
171:
1795:
1611:
1498:
613:
2523:
2513:
2332:
1877:
1740:
1568:
1530:
1387:. Culture and History of Mathematics. Vol. 5. New Delhi: Hindustan Book Agency. pp. 113–143.
1420:
Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth
779:
1558:
663:
635:
Distribution of commodities and money at proportional rates; deriving arithmetic and geometric sums.
421:. From documentary evidence this tomb is known to have been closed in 186 BCE, early in the Western
2237:
2196:
2082:
1847:
1580:
1575:
1542:
1163:
2359:
2343:
2325:
2318:
2293:
1890:
1651:
1153:
769:
1656:
1596:
1538:
1033:
907:
In this light, many scholars of the history of Chinese mathematics compare the significance of
753:
1548:
1840:
1808:
1728:
1715:
528:
509:
284:
751:
Linear problems (in two unknowns) solved using the principle known later in the West as the
2451:
2366:
2134:
1990:
1902:
1745:
1661:
1601:
1410:
783:
303:
299:
287:
as one of the books studied by Ma Xu (馬續). Based on the known dates of his younger brother
401:
8:
1950:
1885:
1781:
1563:
1149:
1038:
806:
311:
2074:
1452:
1357:
1349:
513:
483:
307:
1464:
Straffin, Philip D (1998). "Liu Hui and the First Golden Age of Chinese Mathematics".
1308:
992:
Les neuf chapitres: le classique mathématique de la Chine ancienne et ses commentaires
2458:
2223:
2216:
2207:
2175:
2168:
1955:
1456:
1444:
1396:
1341:
1312:
1282:
995:
977:
586:
585:
of fields of various shapes, such as rectangles, triangles, trapezoids, and circles;
444:
382:
655:
2444:
2105:
1813:
1436:
1388:
1361:
1333:
1304:
1274:
527:
used this title in her overview of the book, as did other mathematicians including
360:
122:
2027:
2062:
2041:
1980:
1406:
1026:
732:
More advanced word problems on proportion, involving work, distances, and rates.
439:
233:
1278:
82:
2352:
2300:
2265:
2148:
2127:
2020:
1835:
1380:
1337:
955:
505:
238:
164:
157:
143:
2497:
2272:
2244:
2230:
2182:
2048:
1448:
1369:
1345:
1316:
987:
944:
867:
498:
491:
473:
410:
2481:
2467:
2109:
2055:
1627:
1392:
1076:
The Second International Conference on History of Mathematics and Astronomy
524:
344:
327:
227:
129:
1071:
911:
on the development of Eastern mathematical traditions to that of Euclid's
318:
greatly assisted the development of ancient mathematics in the regions of
2430:
1960:
1774:
1708:
1622:
1521:
422:
364:
241:
mathematicians, who tended to deduce propositions from an initial set of
218:
180:
1374:"What's so baffling about negative numbers?—a cross-cultural comparison"
1353:
2373:
1975:
1970:
1769:
1440:
930:
has no impact at all on modern mathematics. The style and structure of
386:
352:
79:
76:
2258:
1930:
1897:
1764:
1734:
1695:
1490:
778:
Problems of agricultural yields and the sale of animals that lead to
679:
667:
459:
The title of the book has been translated in a wide variety of ways.
378:
326:. Its influence on mathematical thought in China persisted until the
531:
and Anthony W.-C Lun in their translation of Li Yan and Du Shiran's
472:
With only a slight variation, the Japanese historian of mathematics
359:
165 BCE – d. 142 BCE) and Geng Shouchang (fl. 75 BCE – 49 BCE) (see
2279:
2141:
1965:
1925:
1803:
1758:
1636:
683:
671:
448:
962:
Second Edition, 1919 (possibly copied or paraphrased from Mikami).
915:
on the Western mathematical traditions. However, the influence of
782:, solved by a principle indistinguishable from the modern form of
654:
Finding the diameter or side of a shape given its volume or area.
336:
wrote a detailed commentary in 263. He analyses the procedures of
2251:
2189:
1999:
1940:
1920:
1484:
356:
333:
288:
250:
169:
2006:
1945:
1935:
1690:
969:, Archive for History of Exact Sciences, Springer Verlag, 1994.
687:
675:
659:
363:) with the initial arrangement and commentary on the book, yet
348:
277:
612:
Exchange of commodities at different rates; unit pricing; the
291:(馬融) this places the date of composition no later than 93 CE.
1631:
710:
414:
323:
319:
242:
222:
589:. Liu Hui's commentary includes a method for calculation of
2380:
2286:
582:
395:
868:
Completing of squares and solutions of system of equations
805:
Problems involving the principle known in the West as the
542:
stuck and became the standard English title for the book.
2395:
Questions and Replies between Tang Taizong and Li Weigong
1014:Математика в девяти книгах (Mathematika V Devyati Knigah)
192:
2104:
399:
of the 4th century BCE. This is no longer the case. The
1051:
417:
province. It is among the corpus of texts known as the
1090:
796:
767:
742:
723:
700:
645:
626:
603:
573:
148:
134:
1131:
1129:
1114:
1102:
2056:
Treatise on Cold Injury and Miscellaneous Disorders
1230:
1206:
1169:
1147:
1126:
949:The Development of Mathematics in China and Japan
2495:
283:The title is also mentioned in volume 24 of the
2028:Essential Prescriptions from the Golden Cabinet
1218:
843:
1069:
2090:
1752:Luxuriant Dew of the Spring and Autumn Annals
1506:
1009:, Friedrich Vieweg und Sohn Braunsweig, 1968.
791:
762:
737:
718:
695:
640:
621:
598:
568:
96:
74:
1297:International Journal of Engineering Science
965:Abridged English translation: Lam Lay Yong:
2097:
2083:
1513:
1499:
1070:Patrick Morgan, Daniel (2 November 2018).
616:for solving proportions, using fractions.
53:
2413:The Nine Chapters on the Mathematical Art
2035:The Nine Chapters on the Mathematical Art
974:The Nine Chapters on the Mathematical Art
932:The Nine Chapters on the Mathematical Art
928:The Nine Chapters on the Mathematical Art
917:The Nine Chapters on the Mathematical Art
909:The Nine Chapters on the Mathematical Art
878:The Nine Chapters on the Mathematical Art
874:The Nine Chapters on the Mathematical Art
854:The Nine Chapters on the Mathematical Art
850:The Nine Chapters on the Mathematical Art
837:The Nine Chapters on the Mathematical Art
826:The Nine Chapters on the Mathematical Art
775:The two-sided reference (i.e. Equations)
540:The Nine Chapters on the Mathematical Art
518:The Nine Chapters on the Mathematical Art
274:The Nine Chapters on the Mathematical Art
263:Science and technology of the Han dynasty
214:The Nine Chapters on the Mathematical Art
61:The Nine Chapters on the Mathematical Art
43:The Nine Chapters on the Mathematical Art
1463:
1096:
468:Arithmetical Rules of the Nine Sections.
1417:
1368:
1175:
1159:MacTutor History of Mathematics Archive
1120:
1108:
1057:
937:
454:
14:
2496:
1642:Disasters of the Partisan Prohibitions
1520:
1323:
1294:
1273:, Birkhäuser Basel, pp. 133–155,
1268:
1236:
1212:
1135:
1012:Russian translation: E. I Beriozkina,
926:However, it is dismissive to say that
815:
593:and the approximate value of 3.14159.
533:Chinese Mathematics: A Concise History
32:Mathematical Treatise in Nine Sections
2078:
1494:
954:Highly Abridged English translation:
820:
433:is however much less systematic than
18:Nine Chapters on the Mathematical Art
1617:Second Chinese domination of Vietnam
1194:
1192:
1190:
1188:
1186:
1184:
958:: "Arithmetic in Nine Sections", in
947:: "Arithmetic in Nine Sections", in
545:
1858:Protectorate of the Western Regions
1607:First Chinese domination of Vietnam
1426:
1224:
488:History of Mathematics (Smith 1923)
24:
1429:Educational Studies in Mathematics
1007:Neun Bücher Arithmetischer Technik
848:The geometric figures included in
490:, followed the convention used by
25:
2535:
2388:Three Strategies of Huang Shigong
1478:
1248:
1202:. 地球社编辑部. 1992. pp. 515–531.
1181:
976:, Oxford University Press, 1999.
2475:The Twenty-four Filial Exemplars
1868:Translation of government titles
1672:
1005:German translation: Kurt Vogel,
587:manipulation of vulgar fractions
267:
27:Ancient Chinese mathematics text
2014:Book on Numbers and Computation
1261:
1242:
891:
306:(1777–1855). There is also the
193:
170:
2438:Biographies of Exemplary Women
1831:Three Lords and Nine Ministers
1702:Records of the Grand Historian
1141:
1063:
967:Jiu Zhang Suanshu: An Overview
943:Abridged English translation:
310:given in the treatise for the
149:
135:
97:
75:
13:
1:
1723:Records of the Three Kingdoms
1647:Way of the Five Pecks of Rice
1554:Rebellion of the Seven States
1309:10.1016/s0020-7225(98)00036-6
1253:. 北京: 北京师范大学出版社. p. 118.
1044:
419:Zhangjiashan Han bamboo texts
844:Gou Gu (Pythagorean) Theorem
425:. While its relationship to
7:
2468:School Sayings of Confucius
1422:. Taipei: Caves Books, Ltd.
1279:10.1007/978-3-0348-8599-7_7
1020:
797:
780:systems of linear equations
768:
743:
724:
701:
646:
627:
604:
574:
478:Arithmetic in Nine Sections
108:nine chapters on arithmetic
10:
2540:
2333:Thousand Character Classic
1853:Provinces and commanderies
1741:Eight Immortals of Huainan
1569:War of the Heavenly Horses
1338:10.25162/sudhoff-2013-0017
706:Figuring for construction
632:Proportional distribution
260:
256:
29:
2422:
2404:
2342:
2310:
2206:
2158:
2117:
1989:
1911:
1876:
1790:
1681:
1670:
1559:Han dynasty in Inner Asia
1529:
960:A History of Mathematics,
792:
763:
738:
719:
696:
641:
622:
599:
569:
206:
186:
179:
163:
156:
142:
128:
121:
116:
112:
104:
90:
68:
52:
48:
41:
2238:Etiquette and Ceremonial
2197:Spring and Autumn Annals
1612:Trung sisters' rebellion
1576:Han conquest of Gojoseon
1543:Feast at Swan Goose Gate
1418:Needham, Joseph (1986).
1164:University of St Andrews
231:(202 BCE – 186 BCE) and
70:Traditional Chinese
30:Not to be confused with
2504:Mathematics manuscripts
2360:The Methods of the Sima
2344:Seven Military Classics
2326:Hundred Family Surnames
2319:Three Character Classic
2294:Classic of Filial Piety
1891:Ancient Chinese coinage
1652:Yellow Turban Rebellion
476:shortened the title to
92:Simplified Chinese
2519:Han dynasty literature
2266:Commentary of Gongyang
1913:Science and technology
1657:End of the Han dynasty
1597:Han conquest of Nanyue
1393:10.1287/deca.1090.0176
1034:History of mathematics
1016:, Moscow: GITTL, 1957.
994:. Paris: Dunod, 2004.
754:rule of false position
407:Writings on Reckonings
225:, the first being the
2273:Commentary of Guliang
1841:Three Ducal Ministers
1729:Flying Horse of Gansu
1716:Book of the Later Han
1485:Full text of the book
538:Afterwards, the name
510:Wang Ling (historian)
497:Several years later,
285:Book of the Later Han
261:Further information:
172:/kɨut͡ɕɨɐŋsuɑnʑiuɪt̚/
2452:Four Books for Women
2367:Six Secret Teachings
2135:Doctrine of the Mean
1903:Sino-Roman relations
1746:Mawangdui Silk Texts
1662:Battle of Red Cliffs
1602:Han conquest of Dian
1466:Mathematics Magazine
1303:(12–14): 1339–1359.
1150:Robertson, Edmund F.
938:Notable translations
784:Gaussian elimination
664:extraction of square
651:Reducing dimensions
520:for the first time.
455:Western translations
443:, a mathematics and
304:Carl Friedrich Gauss
300:Gaussian elimination
253:in the 3rd century.
195:/kuʔkjaŋsloːnʔɦljud/
2509:Chinese mathematics
1782:Eastern Han Chinese
1683:Society and culture
1588:Southward expansion
1549:Lü Clan disturbance
1148:O'Connor, John J.;
1039:History of geometry
990:and Shuchun Guo is
816:Major contributions
807:Pythagorean theorem
748:Excess and deficit
729:Equitable taxation
713:of various shapes.
314:. The influence of
312:Pythagorean theorem
150:Chiu Chang Suan-shu
2000:Balanced Discourse
1539:Chu–Han Contention
1441:10.1007/bf01273370
821:Real number system
802:Base and altitude
535:(Li and Du 1987).
514:Jiu Zhang Suan shu
484:David Eugene Smith
466:referred to it as
308:mathematical proof
272:The full title of
2524:2nd-century books
2514:Han dynasty texts
2491:
2490:
2459:Lessons for Women
2259:Commentary of Zuo
2224:Book of Documents
2217:Classic of Poetry
2208:Thirteen Classics
2176:Book of Documents
2169:Classic of Poetry
2072:
2071:
1581:Four Commanderies
1402:978-93-80250-06-9
1288:978-3-0348-9696-2
1200:中國文明史 第三卷 秦漢時代 中冊
1060:, pp. 24–25.
1000:978-2-10-049589-4
813:
812:
552:The Nine Chapters
546:Table of contents
435:The Nine Chapters
427:The Nine Chapters
413:opened a tomb in
389:(78–139) and the
316:The Nine Chapters
296:The Nine Chapters
210:
209:
202:
201:
136:Jiǔ Zhāng Suànshù
123:Standard Mandarin
16:(Redirected from
2531:
2445:Classic of Music
2106:Chinese classics
2099:
2092:
2085:
2076:
2075:
1676:
1515:
1508:
1501:
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1203:
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1085:
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1055:
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795:
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773:
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746:
741:
740:
727:
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721:
704:
699:
698:
649:
644:
643:
630:
625:
624:
609:Millet and rice
607:
602:
601:
592:
579:Bounding fields
577:
572:
571:
557:
556:
554:are as follows:
529:John N. Crossley
391:geometry clauses
361:armillary sphere
198:
197:
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175:
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152:
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138:
137:
114:
113:
100:
99:
86:
85:
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39:
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21:
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2338:
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2202:
2154:
2113:
2103:
2073:
2068:
2063:Zhoubi Suanjing
2042:Huangdi Neijing
2007:Book of Origins
1985:
1981:Zhang Zhongjing
1907:
1872:
1786:
1677:
1668:
1564:Han–Xiongnu War
1525:
1519:
1481:
1476:
1403:
1381:Seshadri, C. S.
1376:
1326:Sudhoffs Archiv
1289:
1264:
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1103:
1095:
1091:
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1068:
1064:
1056:
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1047:
1028:Haidao Suanjing
1023:
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846:
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818:
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774:
747:
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705:
650:
631:
608:
590:
578:
548:
523:Later in 1994,
464:Alexander Wylie
457:
440:Zhoubi Suanjing
276:appears on two
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234:Zhoubi Suanjing
105:Literal meaning
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1479:External links
1477:
1475:
1474:
1461:
1435:(4): 345–357.
1424:
1415:
1401:
1370:Mumford, David
1366:
1332:(2): 199–235.
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1249:吴, 文俊 (1982).
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956:Florian Cajori
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2245:Book of Rites
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2183:Book of Rites
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2160:Five Classics
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1472:(3): 163–181.
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1123:, p. 24.
1122:
1117:
1111:, p. 22.
1110:
1105:
1098:
1097:Straffin 1998
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988:Karine Chemla
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982:0-19-853936-3
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945:Yoshio Mikami
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614:Rule of Three
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499:George Sarton
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385:(d. 23), and
384:
381:(78–37 BCE),
380:
375:
374:Nine Chapters
372:
368:
366:
362:
358:
354:
350:
346:
345:formal proofs
342:
341:Nine Chapters
339:
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268:Original book
264:
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239:ancient Greek
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217:is a Chinese
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2311:San Bai Qian
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2026:
2019:
2012:
2005:
1998:
1757:
1750:
1733:
1721:
1714:
1707:
1700:
1628:Red Eyebrows
1469:
1465:
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1384:
1329:
1325:
1300:
1296:
1270:
1262:Bibliography
1250:
1244:
1232:
1220:
1208:
1199:
1176:Mumford 2010
1171:
1157:
1143:
1121:Needham 1986
1116:
1109:Needham 1986
1104:
1092:
1080:. Retrieved
1075:
1065:
1058:Needham 1986
1053:
1027:
1013:
1006:
991:
973:
966:
959:
948:
931:
927:
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920:
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912:
908:
906:
901:
897:
895:
892:Significance
886:
882:
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873:
871:
862:
858:
853:
849:
847:
836:
835:
831:
825:
824:
752:
551:
550:Contents of
549:
539:
537:
532:
525:Lam Lay Yong
522:
517:
503:
496:
487:
482:
477:
471:
467:
461:
458:
438:
434:
431:Suàn shù shū
430:
426:
406:
402:Suàn shù shū
400:
394:
373:
370:
369:
340:
337:
332:
328:Qing dynasty
315:
295:
293:
282:
273:
271:
247:
232:
228:Suan shu shu
226:
213:
212:
211:
130:Hanyu Pinyin
60:
42:
36:
2431:Bai Hu Tong
2405:Mathematics
1961:Luoxia Hong
1824:Family tree
1709:Book of Han
1623:Xin dynasty
1522:Han dynasty
1237:Dauben 1998
1213:Dauben 1992
1136:Dauben 2013
1082:25 December
709:Volumes of
512:translated
423:Han dynasty
365:Han dynasty
219:mathematics
181:Old Chinese
2498:Categories
2374:Wei Liaozi
2119:Four Books
1976:Zhang Heng
1971:Wang Chong
1792:Government
1770:Han poetry
1045:References
668:cube roots
387:Zhang Heng
353:Zhang Cang
188:Zhengzhang
144:Wade–Giles
59:A page of
2110:Confucian
1951:Liu Xiang
1931:Ding Huan
1898:Silk Road
1765:Old Texts
1735:Huainanzi
1696:Sima Qian
1487:(Chinese)
1457:120420378
1449:0013-1954
1346:0039-4564
1317:0020-7225
1154:"Liu Hui"
896:The word
770:Fangcheng
702:Shanggong
680:perimeter
658:by mixed
647:Shaoguang
563:contents
504:In 1959,
486:, in his
462:In 1852,
445:astronomy
405:(算數書) or
379:Jing Fang
349:Euclidean
2280:Analects
2142:Analects
1966:Sima Tan
1926:Cai Yong
1804:Ban Chao
1796:military
1759:Yiwu Zhi
1637:Chengjia
1372:(2010).
1354:43694474
1225:Siu 1993
1021:See also
921:Elements
913:Elements
744:Yingbuzu
684:diameter
672:diameter
656:Division
575:Fangtian
560:chapter
449:Cai Yong
2301:Mencius
2252:I Ching
2190:I Ching
2149:Mencius
2021:Fangyan
1956:Liu Xin
1941:Hua Tuo
1921:Cai Lun
1886:Coinage
1878:Economy
1814:Emperor
1809:Ma Yuan
1531:History
1411:2648497
1383:(ed.).
1362:1159700
1271:Amphora
1251:九章算术与刘辉
951:, 1913.
660:numbers
393:of the
383:Liu Xin
347:in the
334:Liu Hui
302:before
289:Ma Rong
257:History
251:Liu Hui
2423:Others
1946:Liu An
1936:Du Shi
1691:Ban Gu
1524:topics
1455:
1447:
1409:
1399:
1360:
1352:
1344:
1315:
1285:
998:
980:
725:Junshu
711:solids
688:circle
676:sphere
628:Cuifen
278:bronze
243:axioms
2482:Xunzi
2381:Wu Zi
2112:texts
1991:Texts
1848:Kings
1632:Lulin
1453:S2CID
1379:. In
1377:(PDF)
1358:S2CID
1350:JSTOR
902:zhang
798:Gougu
583:Areas
415:Hubei
330:era.
324:Japan
320:Korea
223:China
2287:Erya
2108:and
1819:list
1794:and
1630:and
1445:ISSN
1397:ISBN
1342:ISSN
1313:ISSN
1283:ISBN
1084:2023
996:ISBN
978:ISBN
682:and
666:and
605:Sumi
508:and
396:Mozi
322:and
98:九章算术
1437:doi
1389:doi
1334:doi
1305:doi
1275:doi
1078:: 4
898:jiu
739:盈不足
686:of
674:of
516:as
371:The
357:fl.
338:The
2500::
1775:Fu
1470:71
1468:.
1451:.
1443:.
1433:24
1431:.
1407:MR
1405:.
1395:.
1356:.
1348:.
1340:.
1330:97
1328:.
1311:.
1301:36
1299:.
1281:,
1183:^
1162:,
1156:,
1152:,
1128:^
1074:.
809:.
793:勾股
786:.
764:方程
757:.
720:均輸
697:商功
690:.
678:,
670:;
662:;
642:少廣
623:衰分
600:粟米
570:方田
494:.
480:.
451:.
245:.
83:算術
2098:e
2091:t
2084:v
1864:)
1860:(
1619:)
1590:(
1545:)
1541:(
1514:e
1507:t
1500:v
1459:.
1439::
1413:.
1391::
1364:.
1336::
1319:.
1307::
1277::
1239:.
1227:.
1215:.
1178:.
1138:.
1086:.
1002:.
591:π
355:(
80:章
77:九
34:.
20:)
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