1274:
614:
755:. After the elimination is complete, one is left to find numerically the real roots of a single-variable equation. Horner's method, though well known in China, was not transmitted to Japan in its final form. So Seki had to work it out by himself independently. He is sometimes credited with Horner's method, which is not historically correct. He also suggested an improvement to Horner's method: to omit higher order terms after some iterations. This practice happens to be the same as that of
1253:
1241:
1265:
455:
509:(古今算法記), in which he gave the first comprehensive account of Chinese algebra in Japan. He successfully applied it to problems suggested by his contemporaries. Before him, these problems were solved using arithmetical methods. In the end of the book, he challenged other mathematicians with 15 new problems, which require multi-variable algebraic equations.
318:
57:
625:
In his book of 1674, however, Seki gave only single-variable equations resulting from elimination, but no account of the process at all, nor his new system of algebraic symbols. There were a few errors in the first edition. A mathematician in
Hashimoto's school criticized the work, saying "only three
750:
With elimination theory in hand, a large part of the problems treated in Seki's time became solvable in principle, given the
Chinese tradition of geometry almost reduced to algebra. In practice, the method could founder under huge computational complexity. Yet this theory had a significant influence
445:
In the
Chinese approach to polynomial interpolation, the motivation was to predict the motion of celestial bodies from observed data. The method was also applied to find various mathematical formulas. Seki learned this technique, most likely, through his close examination of Chinese calendars.
437:
Later, they developed a method that uses two-dimensional arrays, representing four variables at most, but the scope of this method was limited. Accordingly, a target of Seki and his contemporary
Japanese mathematicians was the development of general multivariable algebraic equations and
665:
The effect of the introduction of the new symbolism was not restricted to algebra. With it, mathematicians at that time became able to express mathematical results in more general and abstract way. They concentrated on the study of elimination of variables.
1273:
730:
In comparison with
European mathematics, Seki's first manuscript was as early as Leibniz's first commentary on the subject, which treated matrices only up to the 3x3 case. The subject was forgotten in the West until
351:, they reduced geometric problems to algebra systematically. The number of unknowns in an equation was, however, quite limited. They used notations of an array of numbers to represent a formula; for example,
223:
are Seki's, since many of them appear only in writings of his pupils, some of the results parallel or anticipate those discovered in Europe. For example, he is credited with the discovery of
304:
project to produce a reliable map of his employer's land. He spent many years in studying 13th-century
Chinese calendars to replace the less accurate one used in Japan at that time.
727:
case. The relationships between these works are not clear. Seki developed his mathematics in competition with mathematicians in Osaka and Kyoto, at the cultural center of Japan.
534:. Although it was possible to represent equations of an arbitrary degree (he once treated the 1458th degree) with negative coefficients, there were no symbols corresponding to
432:
387:
644:(算法明記), and gave new solutions to Sawaguchi's 15 problems, using his version of multivariable algebra, similar to Seki's. To answer criticism, in 1685, Takebe Katahiro
332:
as a whole) was based on mathematical knowledge accumulated from the 13th to 15th centuries. The material in these works consisted of algebra with numerical methods,
608:
573:
1756:
1751:
701:
Tanaka came up with the same idea independently. An indication appeared in his book of 1678: some of equations after elimination are the same as resultant. In
1771:
1722:
1558:
Les
Mathematiques Japonaises a L'Epoque d'Edo (1600–1868): Une Etude des Travaux de Seki Takakazu (?-1708) et de Takebe Katahiro (1664–1739).
463:
1240:
1252:
886:
1668:
1644:
1626:
1606:
1586:
1566:
336:
and its applications, and indeterminate integer equations. Seki's work is more or less based on and related to these known methods.
1138:
1029:
770:
for assisting in numerical solution. The most notable of these are the conditions for the existence of multiple roots based on the
243:
1660:
610:. Later, the system was improved by other mathematicians, and in the end it became as expressive as the ones developed in Europe.
1766:
1120:
865:
212:, Seki's work was independent. His successors later developed a school dominant in Japanese mathematics until the end of the
1556:
1091:
983:
881:
286:
912:
902:
1761:
938:
690:. While in his manuscript the formula for 5×5 matrices is obviously wrong, being always 0, in his later publication,
705:(算法紛解) (1690?), he explicitly described the resultant and applied it to several problems. In 1690, Izeki Tomotoki
958:
613:
1727:
1616:
333:
774:, which is the resultant of a polynomial and its "derivative": His working definition of "derivative" was the
759:, but with a completely different perspective. Neither he nor his pupils had, strictly speaking, the idea of
953:
694:(大成算経), written in 1683–1710 with Katahiro Takebe (建部 賢弘) and his brothers, a correct and general formula (
907:
543:
527:
197:
1494:
1685:
479:
344:
201:
1776:
1746:
1353:
1308:
1137:; he obtained a value for π that was correct to the 10th decimal place, using what is now called the
1113:
968:
933:
740:
186:
1650:
196:
He created a new algebraic notation system and, motivated by astronomical computations, did work on
1713:
1313:
1073:
1008:
998:
767:
348:
1677:
1508:
963:
675:
439:
392:
347:
in the 19th century) of arbitrary-degree algebraic equation with real coefficients. By using the
97:
1173:
encompasses roughly 50+ works in 50+ publications in three languages and 100+ library holdings.
756:
340:
354:
1717:
1664:
1640:
1622:
1602:
1582:
1562:
1437:
1434:
1416:
1154:
1133:
Another of Seki's contributions was the rectification of the circle, i.e., the calculation of
744:
695:
224:
205:
1397:
1394:
1374:
1371:
1596:
1576:
1142:
855:
791:
578:
270:
247:
1293:, the custom of presenting mathematical problems, carved in wood tablets, to the public in
1652:
Encyclopaedia of the
History of Science, Technology, and Medicine in Non-Western Cultures.
1552:
1337:
Encyclopaedia of the
History of Science, Technology, and Medicine in Non-Western Cultures,
1106:
1083:
1052:
1035:
1013:
549:
265:
Not much is known about Seki's personal life. His birthplace has been indicated as either
1694:
735:
in 1750 was brought to it by the same motivations. Elimination theory equivalent to the
1465:
1294:
993:
732:
266:
81:
1225:
1208:
1190:
1740:
1681:
1512:
1332:
1078:
860:
322:
175:
62:
31:
235:(the first in 1683, the complete version no later than 1710) are attributed to him.
1444:
1421:
1403:
1381:
1358:
1264:
978:
797:
He obtained some evaluations of the number of real roots of a polynomial equation.
771:
662:, in which he showed in detail the process of elimination using algebraic symbols.
209:
204:. Although he was a contemporary of German polymath mathematician and philosopher
1003:
988:
943:
917:
809:
719:(算法発揮), in which he gave resultant and Laplace's formula of determinant for the
687:
539:
232:
137:
1477:
760:
516:(発微算法), giving solutions to all the 15 problems. The method he used is called
254:
213:
179:
1165:
In a statistical overview derived from writings by and about Seki
Takakazu,
948:
679:
454:
301:
228:
715:, a mathematician active in Osaka but not in Hashimoto's school, published
292:
1672:
1570:
1547:
1630:
1610:
1590:
1481:
1472:
June 25, 2008. Seki was greatly influenced by Chinese mathematical books
1170:
531:
281:
17:
1691:
1299:
1290:
535:
35:
317:
242:
correct to the 10th decimal place, having used what is now called the
1656:
1303:
1057:
1706:
686:(解伏題之法). To express the resultant, he developed the notion of the
637:
612:
522:
502:
471:
467:
453:
316:
103:
85:
1484:. (とくに大きな影響を受けたのは、中国から伝わった数学書『算学啓蒙』(1299年)と『楊輝算法』(1274-75年)だった。)
1166:
56:
1637:
Kinsei Nihon Suugakushi -Seki Takakazu no jitsuzou wo motomete.
621:(1712), tabulating binomial coefficients and Bernoulli numbers
274:
253:
Seki was influenced by Japanese mathematics books such as the
77:
1466:和算の開祖 関孝和 ("Seki Takakazu, founder of Japanese mathematics")
1618:
Mathematics in Society and History: Sociological Inquiries.
1398:
Mathematics in Society and History: Sociological Inquiries,
1134:
818:
239:
185:
Seki laid foundations for the subsequent development of
1246:
Seki on a 1992 stamp, taken from an Edo era ink drawing
321:
Ink drawing of Seki Takakazu, from the archives of the
219:
While it is not clear how much of the achievements of
1279:
Seki's grave marker outside Jyōrin-ji temple in Tokyo
581:
552:
395:
357:
339:
Chinese algebraists discovered numerical evaluation (
290:, and adopted into the Seki family, a subject of the
636:, who was from Hashimoto's school and was active in
133:
118:
110:
92:
70:
47:
602:
567:
426:
381:
626:out of 15 are correct." In 1678, Tanaka Yoshizane
208:and British polymath physicist and mathematician
1695:Alternate online, full-text copy at archive.org
1539:
1578:An Introduction to the History of Mathematics.
1533:
1496:An Introduction to the History of Mathematics,
1218:
1201:
1183:
708:
647:
629:
494:
484:
167:
154:
1114:
193:. He has been described as "Japan's Newton".
8:
277:. His birth date ranges from 1635 to 1643.
1601:Belmont, California: Thomson Brooks/Cole.
1347:
1345:
1121:
1107:
804:
44:
580:
551:
403:
394:
356:
1561:Paris: Librairie Philosophique J. Vrin.
1456:鳴海風「和算」『東京人』第321号、都市出版、52-56頁、2013年2月3日。
61:Ink painting of Seki Takakazu, from the
1723:MacTutor History of Mathematics Archive
1621:Dordrecht: Kluwer Academic Publishers.
1325:
1258:Memorial to Seki, with stele and statue
1236:
1065:
1044:
1021:
925:
894:
873:
847:
816:
1598:Linear algebra: a Modern Introduction.
1474:Introduction to Computational Studies
1438:Linear algebra: a Modern Introduction,
1141:, rediscovered in the 20th century by
973:
747:was established no earlier than 1750.
464:National Museum of Nature and Science
7:
1757:18th-century Japanese mathematicians
1752:17th-century Japanese mathematicians
766:Seki also studied the properties of
751:on the direction of development of
159:, c. March 1642 – December 5, 1708)
1772:Japanese writers of the Edo period
1687:A History of Japanese Mathematics.
1639:Tokyo: University of Tokyo Press.
1480:and Yang Hui suan fa (1274-75) by
1375:A History of Japanese Mathematics,
654:, one of Seki's pupils, published
238:Seki also calculated the value of
25:
1690:Chicago: Open Court Publishing.
1272:
1263:
1251:
1239:
674:In 1683, Seki pushed ahead with
55:
1529:History of mathematics in Japan
491:, a pupil of Hashimoto Masakazu
1575:Howard Whitley, Eves. (1990).
1157:is named after Seki Takakazu.
1139:Aitken's delta-squared process
698:for the determinant) appears.
376:
358:
244:Aitken's delta-squared process
1:
790:), which was computed by the
450:Competing with contemporaries
1540:
520:. He introduced the use of
427:{\displaystyle ax^{2}+bx+c}
1793:
526:to represent unknowns and
313:Chinese mathematical roots
284:clan, a subject of Ko-shu
29:
1534:
1219:
1202:
1184:
913:Madhava's correction term
739:form was rediscovered by
709:
648:
630:
495:
485:
382:{\displaystyle (a\ b\ c)}
168:
155:
143:
126:
54:
1728:University of St Andrews
1718:"Takakazu Shinsuke Seki"
1649:Selin, Helaine. (1997).
1615:Restivo, Sal P. (1992).
1581:Philadelphia: Saunders.
1392:Restivo, Sal P. (1992).
1092:Other topics related to
512:In 1674, Seki published
334:polynomial interpolation
246:, rediscovered later by
1767:Japanese mathematicians
1635:Sato, Kenichi. (2005),
1527:Endō Toshisada (1896).
300:, he was involved in a
1595:Poole, David. (2005).
1493:Eves, Howard. (1990).
1432:Poole, David. (2005).
817:mathematical constant
622:
604:
603:{\displaystyle ax+b=0}
569:
475:
428:
383:
325:
198:infinitesimal calculus
27:Japanese mathematician
1369:Smith, David. (1914)
866:Use in other formulae
757:Newton–Raphson method
616:
605:
570:
457:
429:
384:
345:William George Horner
328:His mathematics (and
320:
202:Diophantine equations
1714:Robertson, Edmund F.
1309:Japanese mathematics
1214:Seki Takakazu Zenshū
810:a series of articles
656:Hatsubi Sanpō Genkai
579:
568:{\displaystyle ax+b}
550:
393:
355:
343:, re-established by
187:Japanese mathematics
1712:O'Connor, John J.;
1509:WorldCat Identities
1314:Napkin ring problem
1074:Squaring the circle
1009:Chudnovsky brothers
999:Srinivasa Ramanujan
768:algebraic equations
658:(発微算法諺解), notes on
617:A page from Seki's
349:Pythagorean theorem
280:He was born to the
1762:Linear algebraists
1678:David Eugene Smith
1541:Dai Nihon sūgakush
964:Ludolph van Ceulen
676:elimination theory
670:Elimination theory
623:
600:
565:
480:Sawaguchi Kazuyuki
476:
440:elimination theory
424:
379:
326:
296:. While in Ko-shu
178:and author of the
98:Gregorian calendar
96:December 5, 1708 (
65:archives in Tokyo.
1513:関孝和 ca. 1642-1708
1228:, collected works
1155:7483 Sekitakakazu
1131:
1130:
801:Calculation of pi
745:Laplace's formula
696:Laplace's formula
462:exhibited in the
372:
366:
225:Bernoulli numbers
206:Gottfried Leibniz
174:, was a Japanese
147:
146:
128:Scientific career
16:(Redirected from
1784:
1730:
1553:Horiuchi, Annick
1546:. Tōkyō: _____.
1545:
1543:
1537:
1536:
1515:
1506:
1500:
1491:
1485:
1463:
1457:
1454:
1448:
1447:; Selin, p. 891.
1430:
1424:
1412:
1406:
1390:
1384:
1367:
1361:
1349:
1340:
1330:
1276:
1267:
1255:
1243:
1224:
1222:
1221:
1207:
1205:
1204:
1189:
1187:
1186:
1143:Alexander Aitken
1123:
1116:
1109:
1095:
1087:
959:Jamshīd al-Kāshī
856:Area of a circle
842:
841:
838:
835:
832:
821:
805:
792:binomial theorem
714:
712:
711:
684:Kaifukudai no Hō
653:
651:
650:
635:
633:
632:
609:
607:
606:
601:
575:could also mean
574:
572:
571:
566:
500:
498:
497:
490:
488:
487:
433:
431:
430:
425:
408:
407:
388:
386:
385:
380:
370:
364:
271:Gunma Prefecture
248:Alexander Aitken
173:
171:
170:
161:, also known as
160:
158:
157:
119:Other names
59:
45:
21:
1792:
1791:
1787:
1786:
1785:
1783:
1782:
1781:
1737:
1736:
1734:
1711:
1703:
1531:
1524:
1519:
1518:
1507:
1503:
1492:
1488:
1464:
1460:
1455:
1451:
1431:
1427:
1413:
1409:
1391:
1387:
1368:
1364:
1350:
1343:
1331:
1327:
1322:
1287:
1280:
1277:
1268:
1259:
1256:
1247:
1244:
1235:
1216:
1199:
1181:
1163:
1151:
1127:
1093:
1085:
1053:Indiana pi bill
1036:A History of Pi
1014:Yasumasa Kanada
839:
836:
833:
830:
828:
819:
803:
706:
672:
645:
627:
577:
576:
548:
547:
546:. For example,
492:
482:
452:
399:
391:
390:
353:
352:
341:Horner's method
315:
310:
263:
165:
152:
106:
101:
88:
75:
66:
50:
43:
28:
23:
22:
15:
12:
11:
5:
1790:
1788:
1780:
1779:
1774:
1769:
1764:
1759:
1754:
1749:
1739:
1738:
1732:
1731:
1709:
1702:
1701:External links
1699:
1698:
1697:
1675:
1673:OCLC 186451909
1647:
1633:
1613:
1593:
1573:
1571:OCLC 318334322
1550:
1548:OCLC 122770600
1523:
1520:
1517:
1516:
1501:
1486:
1470:Otonanokagaku.
1458:
1449:
1425:
1407:
1385:
1362:
1341:
1333:Selin, Helaine
1324:
1323:
1321:
1318:
1317:
1316:
1311:
1306:
1297:
1295:Shinto shrines
1286:
1283:
1282:
1281:
1278:
1271:
1269:
1262:
1260:
1257:
1250:
1248:
1245:
1238:
1234:
1231:
1230:
1229:
1226:OCLC 006343391
1211:
1209:OCLC 049703813
1193:
1191:OCLC 045626660
1162:
1161:Selected works
1159:
1150:
1147:
1129:
1128:
1126:
1125:
1118:
1111:
1103:
1100:
1099:
1098:
1097:
1089:
1081:
1076:
1068:
1067:
1066:Related topics
1063:
1062:
1061:
1060:
1055:
1047:
1046:
1042:
1041:
1040:
1039:
1032:
1024:
1023:
1019:
1018:
1017:
1016:
1011:
1006:
1001:
996:
994:William Shanks
991:
986:
981:
976:
971:
969:François Viète
966:
961:
956:
951:
946:
941:
936:
928:
927:
923:
922:
921:
920:
915:
910:
908:Approximations
905:
903:Less than 22/7
897:
896:
892:
891:
890:
889:
884:
876:
875:
871:
870:
869:
868:
863:
858:
850:
849:
845:
844:
824:
823:
814:
813:
802:
799:
741:Étienne Bézout
733:Gabriel Cramer
671:
668:
599:
596:
593:
590:
587:
584:
564:
561:
558:
555:
507:Kokon Sanpō Ki
451:
448:
423:
420:
417:
414:
411:
406:
402:
398:
378:
375:
369:
363:
360:
314:
311:
309:
306:
262:
259:
145:
144:
141:
140:
135:
131:
130:
124:
123:
120:
116:
115:
112:
108:
107:
102:
94:
90:
89:
76:
72:
68:
67:
60:
52:
51:
48:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1789:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1744:
1742:
1735:
1729:
1725:
1724:
1719:
1715:
1710:
1708:
1705:
1704:
1700:
1696:
1693:
1689:
1688:
1683:
1682:Yoshio Mikami
1679:
1676:
1674:
1670:
1669:9780792340669
1666:
1662:
1658:
1654:
1653:
1648:
1646:
1645:4-13-061355-3
1642:
1638:
1634:
1632:
1631:OCLC 25709270
1628:
1627:9780792317654
1624:
1620:
1619:
1614:
1612:
1611:OCLC 67379937
1608:
1607:9780534998455
1604:
1600:
1599:
1594:
1592:
1591:OCLC 20842510
1588:
1587:9780030295584
1584:
1580:
1579:
1574:
1572:
1568:
1567:9782711612130
1564:
1560:
1559:
1554:
1551:
1549:
1542:
1530:
1526:
1525:
1521:
1514:
1510:
1505:
1502:
1499:
1497:
1490:
1487:
1483:
1479:
1475:
1471:
1467:
1462:
1459:
1453:
1450:
1446:
1443:, p. 279, at
1442:
1441:
1436:
1435:
1429:
1426:
1423:
1420:, p. 128, at
1419:
1418:
1417:pp. 128-142.
1411:
1408:
1405:
1401:
1400:
1396:
1395:
1389:
1386:
1383:
1379:
1378:
1373:
1372:
1366:
1363:
1360:
1357:, p. 641, at
1356:
1355:
1348:
1346:
1342:
1338:
1334:
1329:
1326:
1319:
1315:
1312:
1310:
1307:
1305:
1302:, a Japanese
1301:
1298:
1296:
1292:
1289:
1288:
1284:
1275:
1270:
1266:
1261:
1254:
1249:
1242:
1237:
1232:
1227:
1215:
1212:
1210:
1198:
1197:Katsuyō Sanpō
1194:
1192:
1180:
1176:
1175:
1174:
1172:
1168:
1160:
1158:
1156:
1153:The asteroid
1148:
1146:
1144:
1140:
1136:
1124:
1119:
1117:
1112:
1110:
1105:
1104:
1102:
1101:
1096:
1090:
1088:
1084:Six nines in
1082:
1080:
1079:Basel problem
1077:
1075:
1072:
1071:
1070:
1069:
1064:
1059:
1056:
1054:
1051:
1050:
1049:
1048:
1043:
1038:
1037:
1033:
1031:
1028:
1027:
1026:
1025:
1020:
1015:
1012:
1010:
1007:
1005:
1002:
1000:
997:
995:
992:
990:
987:
985:
984:William Jones
982:
980:
977:
975:
974:Seki Takakazu
972:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
940:
937:
935:
932:
931:
930:
929:
924:
919:
916:
914:
911:
909:
906:
904:
901:
900:
899:
898:
893:
888:
887:Transcendence
885:
883:
882:Irrationality
880:
879:
878:
877:
872:
867:
864:
862:
861:Circumference
859:
857:
854:
853:
852:
851:
846:
843:
826:
825:
822:
815:
811:
807:
806:
800:
798:
795:
793:
789:
785:
781:
777:
773:
769:
764:
762:
758:
754:
748:
746:
742:
738:
734:
728:
726:
722:
718:
704:
699:
697:
693:
692:Taisei Sankei
689:
685:
681:
677:
669:
667:
663:
661:
660:Hatsubi Sanpō
657:
643:
639:
620:
619:Katsuyō Sanpō
615:
611:
597:
594:
591:
588:
585:
582:
562:
559:
556:
553:
545:
541:
537:
533:
529:
525:
524:
519:
515:
514:Hatsubi Sanpō
510:
508:
504:
481:
473:
469:
465:
461:
460:Hatsubi Sanpō
456:
449:
447:
443:
441:
435:
421:
418:
415:
412:
409:
404:
400:
396:
373:
367:
361:
350:
346:
342:
337:
335:
331:
324:
323:Ishikawa clan
319:
312:
307:
305:
303:
299:
295:
294:
289:
288:
283:
278:
276:
272:
268:
260:
258:
256:
251:
249:
245:
241:
236:
234:
230:
226:
222:
217:
215:
211:
207:
203:
199:
194:
192:
188:
183:
181:
177:
176:mathematician
164:
151:
150:Seki Takakazu
142:
139:
136:
132:
129:
125:
121:
117:
113:
109:
105:
99:
95:
91:
87:
83:
79:
73:
69:
64:
63:Japan Academy
58:
53:
49:Seki Takakazu
46:
41:
37:
33:
32:Japanese name
19:
1733:
1721:
1707:Sugaku-bunka
1692:OCLC 1515528
1686:
1651:
1636:
1617:
1597:
1577:
1557:
1528:
1504:
1495:
1489:
1473:
1469:
1461:
1452:
1445:Google Books
1439:
1433:
1428:
1422:Google Books
1415:
1410:
1404:Google Books
1402:, p. 56, at
1399:
1393:
1388:
1382:Google Books
1380:, p. 91, at
1377:pp. 91-127.
1376:
1370:
1365:
1359:Google Books
1352:
1336:
1328:
1213:
1196:
1178:
1164:
1152:
1132:
1034:
979:Takebe Kenko
918:Memorization
827:
796:
787:
783:
779:
775:
772:discriminant
765:
752:
749:
736:
729:
724:
720:
716:
703:Sanpō Funkai
702:
700:
691:
683:
673:
664:
659:
655:
641:
624:
618:
521:
517:
513:
511:
506:
505:, published
477:
459:
444:
436:
338:
329:
327:
297:
291:
285:
279:
264:
252:
237:
220:
218:
210:Isaac Newton
195:
190:
184:
162:
149:
148:
127:
39:
1777:1642 births
1747:1708 deaths
1655:Dordrecht:
1179:Kenpu no Hō
1004:John Wrench
989:John Machin
944:Zu Chongzhi
717:Sanpō Hakki
688:determinant
678:, based on
642:Sanpō Meiki
640:, authored
536:parentheses
458:Replica of
233:determinant
189:, known as
138:Mathematics
111:Nationality
1741:Categories
1684:. (1914).
1555:. (1994).
1522:References
1478:Zhu Shijie
1476:(1299) by
1335:. (1997).
1045:In culture
1030:Chronology
934:Archimedes
874:Properties
761:derivative
680:resultants
214:Edo period
180:Edo period
949:Aryabhata
778:-term in
743:in 1764.
682:, in the
532:equations
528:variables
478:In 1671,
302:surveying
261:Biography
229:resultant
163:Seki Kōwa
122:Seki Kōwa
18:Seki Kowa
1661:Springer
1482:Yang Hui
1440:p. 279.
1285:See also
1171:WorldCat
840:26433...
808:Part of
544:division
540:equality
518:bōsho-hō
282:Uchiyama
114:Japanese
30:In this
1498:p. 405.
1414:Smith,
1354:p. 641.
1351:Selin,
1300:Soroban
1291:Sangaku
1233:Gallery
1195:1712 –
1177:1683 –
1022:History
954:Madhava
939:Liu Hui
829:3.14159
267:Fujioka
255:Jinkōki
82:Fujioka
74:1642(?)
36:surname
1667:
1657:Kluwer
1643:
1625:
1605:
1585:
1565:
1535:日本數學史史
1339:p. 890
1304:abacus
1149:Legacy
1058:Pi Day
926:People
812:on the
371:
365:
308:Career
293:shōgun
227:. The
134:Fields
34:, the
1320:Notes
1220:關孝和全集
895:Value
837:23846
834:89793
831:26535
753:wasan
737:wasan
710:井関 知辰
649:建部 賢弘
638:Kyoto
631:田中 由真
542:, or
523:kanji
503:Osaka
496:橋本 正数
486:沢口 一之
472:Japan
468:Tokyo
330:wasan
273:, or
221:wasan
191:wasan
104:Japan
86:Japan
1680:and
1665:ISBN
1641:ISBN
1623:ISBN
1603:ISBN
1583:ISBN
1563:ISBN
1203:括要算法
1185:驗符之法
1167:OCLC
848:Uses
776:O(h)
389:for
231:and
200:and
169:関 孝和
156:関 孝和
93:Died
71:Born
40:Seki
530:in
501:in
298:han
287:han
275:Edo
269:in
80:or
78:Edo
38:is
1743::
1726:,
1720:,
1716:,
1671:;
1663:.
1629:;
1609:;
1589:;
1569:;
1538:,
1511::
1468:,
1344:^
1145:.
1135:pi
794:.
786:+
763:.
538:,
470:,
466:,
442:.
434:.
257:.
250:.
240:pi
216:.
182:.
84:,
1659:/
1544:)
1532:(
1223:)
1217:(
1206:)
1200:(
1188:)
1182:(
1169:/
1122:e
1115:t
1108:v
1094:π
1086:π
820:π
788:h
784:x
782:(
780:f
725:n
723:×
721:n
713:)
707:(
652:)
646:(
634:)
628:(
598:0
595:=
592:b
589:+
586:x
583:a
563:b
560:+
557:x
554:a
499:)
493:(
489:)
483:(
474:.
422:c
419:+
416:x
413:b
410:+
405:2
401:x
397:a
377:)
374:c
368:b
362:a
359:(
172:)
166:(
153:(
100:)
42:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
