413:
496:"If the finding that samples of the same manuscript would be centuries apart is not based on mistakes ... there are still some factors that have evidently been overlooked by the Bodleian research team: the well-known divergence in exposure to cosmic radiation at different altitudes and the possible variation in background radiation due to the presence of certain minerals in exposed, mountainous rock have nowhere been taken into account."
313:, have not been identified precisely. It is probable that most of the rules and examples had been originally composed in Sanskrit, while one of the sections was written entirely in a dialect. It is possible that the manuscript might be a compilation of fragments from different works composed in a number of language varieties. Hayashi admits that some of the irregularities are due to errors by scribes or may be orthographical.
50:
237:
481:
have permitted serious collegial discussion and peer review prior to public announcements. While the excitement inspired by intriguing discoveries benefits our field and scholarly research in general, the confusion generated by broadcasting over-eager and carelessly inferred conclusions, with their inevitable aftermath of caveats and disputes, does not.
514:
necessary to "take quite seriously the judgement of palaeographists such as
Richard Salomon who observed that, what he teleologically called āProto-ÅÄradÄ,ā āfirst emerged around the middle of the seventh centuryā (Salomon 1998: 40). This excludes the earlier dates attributed to manuscript folios on which a fully developed form of ÅÄradÄ appears."
181:". For some portions a carbon-date was proposed of AD 224ā383 while for other portions a carbon-date as late as AD 885ā993 in a recent study, but the dating has been criticised by specialists on methodological grounds (Plofker et al. 2017 and Houben 2018 Ā§3). The manuscript contains the earliest known Indian use of a
480:
We express regret that the
Bodleian Library kept their carbon-dating findings embargoed for many months, and then chose a newspaper press-release and YouTube as media for a first communication of these technical and historical matters. The Library thus bypassed standard academic channels that would
501:
Prior to the proposed radiocarbon dates of the 2017 study, most scholars agreed that the physical manuscript was a copy of a more ancient text, whose date had to be estimated partly on the basis of its content. HoernlƩ thought that the manuscript was from the 9th century, but the original was from
244:
The manuscript is a compendium of rules and illustrative examples. Each example is stated as a problem, the solution is described, and it is verified that the problem has been solved. The sample problems are in verse and the commentary is in prose associated with calculations. The problems involve
513:
To settle the date of the
Bakhshali manuscript, language use and especially palaeography are other major parameters to be taken into account. In this context Houben observed: "it is difficult to derive a linear chronological difference from the observed linguistic variation," and therefore it is
625:
Jan E.M. Houben "Linguistic
Paradox and Diglossia: on the emergence of Sanskrit and Sanskritic language in Ancient India." De Gruyter Open Linguistics (Topical Issue on Historical Sociolinguistic Philology, ed. by Chiara Barbati and Christian Gastgeber.) OPLI ā Vol. 4, issue 1: 1ā18. DOI:
377:) of the example's numerical information in tabular form, then a computation that works out the example by following the rule step-by-step while quoting it, and finally a verification to confirm that the solution satisfies the problem. This is a style similar to that of
502:
the 3rd or 4th century. Indian scholars assigned it an earlier date. Datta assigned it to the "early centuries of the
Christian era". Channabasappa dated it to AD 200ā400, on the grounds that it uses mathematical terminology different from that of
578:
G. R. Kaye, on the other hand, thought in 1927 that the work was composed in the 12th century, but this was discounted in recent scholarship. G. G. Joseph wrote, "It is particularly unfortunate that Kaye is still quoted as an authority on Indian
368:
The manuscript is a compilation of mathematical rules and examples (in verse), and prose commentaries on these verses. Typically, a rule is given, with one or more examples, where each example is followed by a "statement"
996:
301:, was significantly influenced in its phonetics and morphology by a local artist dialect or dialects, and some of the resultant linguistic peculiarities of the text are shared with
2089:
1302:
Plofker, Kim, Agathe Keller, Takao
Hayashi, Clemency Montelle, and Dominik Wujastyk. 2017. "The BakhshÄlÄ« Manuscript: A Response to the Bodleian Libraryās Radiocarbon Datingā.
1004:
1639:
851:
396:
and techniques for a variety of problems, such as systems of linear equations, quadratic equations, arithmetic progressions and arithmetico-geometric series, computing
606:, and Dominik Wujastyk. 2017. "The BakhshÄlÄ« Manuscript: A Response to the Bodleian Libraryās Radiocarbon Dating.ā History of Science in South Asia, 5.1: 134ā150.
1335:
2069:
709:
1284:
1091:
749:
1856:
2031:
2115:
2084:
1279:
1087:
953:
2125:
2094:
1202:
420:
The
Bakhshali manuscript uses numerals with a place-value system, using a dot as a place holder for zero. The dot symbol came to be called the
1159:
1126:
893:
336:. The brahmin might have been the author of the commentary as well as the scribe of the manuscript. Near the colophon appears a broken word
2120:
561:, p. 11), or as the so-called GÄthÄ dialect, the literary form of the Northwestern Prakrit, which combined elements of Sanskrit and
293:, a script which is known for having been in use mainly from the 8th to the 12th century in the northwestern part of South Asia, such as
1609:
2079:
1589:
982:
793:
690:
528:
783:
2135:
1328:
1829:
1679:
1250:
2064:
1433:
71:
AD 224ā383/ 885ā993 (proposed carbon-dates, recently disputed on methodological grounds: Plofker et al. 2017, Houben 2018 Ā§3)
1473:
1290:
476:
and
Dominik Wujastyk to publicly object to the library making the dates globally available, usurping academic precedence:
1352:
270:
2054:
2130:
2074:
2059:
1321:
883:
491:
Referring to the detailed reconsideration of the evidence by
Plofker et al., Sanskrit scholar, Jan Houben remarked:
1569:
1937:
1902:
1579:
258:
1231:
461:). If the dates are valid, it is not known how folios from different centuries came to be collected and buried.
1869:
1584:
1275:
1083:
302:
1995:
1877:
1468:
278:
1176:
1027:
607:
2036:
2016:
1943:
1756:
262:
1873:
1865:
1367:
523:
386:
317:
266:
217:. After his death, it was examined by G.R.Kaye, who edited the work and published it as a book in 1927.
486:
Plofker et al., The BakhshÄlÄ« Manuscript: A Response to the
Bodleian Libraryās Radiocarbon Dating, 2017
1798:
1721:
1458:
1026:
Plofker, Kim; Keller, Agathe; Hayashi, Takao; Montelle, Clemency; Wujastyk, Dominik (6 October 2017).
1955:
1634:
565:
and whose use as a literary language predated the adoption of Classical Sanskrit for this purposely.(
412:
240:
The numerals used in the Bakhshali manuscript, dated to sometime between the 3rd and 7th century AD.
2021:
1990:
1914:
1784:
1614:
1559:
1554:
1508:
274:
214:
154:
1258:
1523:
1344:
997:"Oxford Radiocarbon Accelerator Unit dates the world's oldest recorded origin of the zero symbol"
533:
446:
228:
at the University of Oxford (MS. Sansk. d. 14), though folio are periodically loaned to museums.
210:
178:
1211:
750:"Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'"
674:
1961:
306:
2026:
1908:
1728:
1503:
1498:
1372:
1172:
1155:
1122:
1049:
978:
889:
789:
717:
686:
603:
473:
433:
Prior to the 2017 carbon dating, a 9th-century inscription of zero on the wall of a temple in
310:
1707:
1742:
1624:
1594:
1564:
1478:
1254:
1184:
1136:
1039:
458:
225:
40:
1259:"A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics"
424:(literally, the dot of the empty place). References to the concept are found in Subandhu's
1949:
1896:
1824:
1770:
1619:
1488:
830:"Book Review: G. R. Kaye, The BakhshĆ¢lĆ® ManuscriptāA Study in Mediaeval Mathematics, 1927"
401:
1735:
1528:
1448:
1377:
1140:
949:
683:
Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
345:
1920:
1840:
1777:
1700:
1686:
1604:
1599:
1533:
1428:
450:
290:
125:
1387:
2109:
2000:
1932:
1861:
1763:
1423:
782:
John Newsome Crossley; Anthony Wah-Cheung Lun; Kangshen Shen; Shen Kangsheng (1999).
678:
538:
469:
378:
198:
1513:
236:
49:
1967:
1805:
1749:
1693:
1493:
1418:
1028:"The BakhshÄlÄ« Manuscript: A Response to the Bodleian Library's Radiocarbon Dating"
17:
829:
1177:
The BakhshÄlÄ« Manuscript: A Response to the Bodleian Libraryās Radiocarbon Dating
464:
The publication of the radio carbon dates, initially via non-academic media, led
437:, Madhya Pradesh, was once thought to be the oldest Indian use of a zero symbol.
1973:
1926:
1714:
1660:
1443:
1397:
1168:
879:
465:
454:
426:
397:
297:
and neighbouring regions. The language of the manuscript, though intended to be
638:
All the pages have been photographed which are available in the book by Hayashi
1672:
1629:
1483:
1463:
1438:
1296:
507:
389:, including the emphasis on verification that became obsolete in later works.
350:
246:
221:
1053:
721:
1791:
1453:
1413:
503:
393:
202:
158:
79:
710:"Much ado about nothing: ancient Indian text contains earliest zero symbol"
1574:
1538:
1518:
1307:
627:
510:'s work (AD 629), and said that it was "not much later than Bhaskara I".
356:
298:
254:
186:
174:
170:
166:
90:
608:
https://journals.library.ualberta.ca/hssa/index.php/hssa/article/view/22
360:
etc.), the supposed place where the manuscript might have been written.
1392:
562:
434:
321:
294:
250:
430:, which has been dated between 385 and 465 by the scholar Maan Singh.
1382:
1313:
404:(profit and loss), measurement such as of the fineness of gold, etc.
220:
The extant manuscript is incomplete, consisting of seventy leaves of
206:
162:
1297:"A Big Zero: Research uncovers the date of the Bakhshali Manuscript"
1188:
1044:
1145:, Vienna: Alfred Hƶlder (Editor of the Court and of the University)
785:
The Nine Chapters on the Mathematical Art: Companion and Commentary
1204:
The Bakhshali Manuscript: An Ancient Treatise of Indian Arithmetic
411:
235:
1119:
The BakhshÄlÄ« manuscript: an ancient Indian mathematical treatise
1640:
Shanti Swarup Bhatnagar Prize recipients in Mathematical Science
1210:, Allahabad: Dr. Ratna Kumari Svadhyaya Sansthan, archived from
182:
1317:
598:
596:
1310:
Challenges the claims made in the YouTube video "A Big Zero."
506:. Hayashi noted some similarities between the manuscript and
213:, Pakistan. The first research on the manuscript was done by
1287:(Ian G. Pearce, Indian Mathematics: Redressing the balance)
1069:
The Crest of the Peacock, non-European roots of Mathematics
449:
to three different centuries and empires, from AD 224ā383 (
1841:
Infinite series expansions for the trigonometric functions
1201:
Sarasvati, Svami Satya Prakash; Jyotishmati, Usha (1979),
1152:
The BakhshÄlÄ« manuscripts: a study in medieval mathematics
1835:
1121:. Groningen Oriental studies. Groningen: Egbert Forsten.
852:"London museum showcases India's contribution to science"
305:. The overlying dialects, though sharing affinities with
1232:"On the square root formula in the Bakhshali manuscript"
906:
904:
557:
Variously described either as an "irregular Sanskrit" (
320:
to one of the sections states that it was written by a
224:, whose intended order is not known. It is kept at the
261:. The topics covered include fractions, square roots,
2090:
Ramanujan Institute for Advanced Study in Mathematics
201:
from a field in 1881, by a peasant in the village of
2045:
2009:
1983:
1887:
1849:
1815:
1652:
1547:
1406:
1360:
1351:
416:
Bakhshali manuscript, detail of the numeral "zero".
139:
131:
121:
113:
105:
97:
86:
75:
67:
59:
34:
744:
742:
740:
738:
445:In 2017, samples from 3 folios of the corpus were
177:). It is perhaps "the oldest extant manuscript in
153:is an ancient Indian mathematical text written on
1082:J. J. O'Connor; E. F. Robertson (November 2000).
948:J. J. O'Connor; E. F. Robertson (November 2000).
289:The manuscript is written in an earlier form of
493:
478:
1071:, Princeton University Press, pp. 215ā216
777:
775:
773:
771:
668:
666:
664:
1329:
823:
821:
819:
817:
662:
660:
658:
656:
654:
652:
650:
648:
646:
644:
8:
602:Plofker, Kim, Agathe Keller, Takao Hayashi,
185:symbol. It is written in a form of literary
1226:with complete text in Devanagari, 110 pages
1001:School of Archaeology, University of Oxford
922:Section VII 11, corresponding to folio 46.(
888:, Princeton University Press, p. 158,
348:as being in northwestern India (along with
328:", a "king of calculators," for the use of
1857:Kerala school of astronomy and mathematics
1357:
1336:
1322:
1314:
1291:Hoernle: On the Bakhshali Manuscript, 1887
874:
872:
340:, which has been interpreted as the place
31:
1043:
685:, vol. 1, Springer, pp. B1āB3,
621:
619:
617:
615:
157:that was found in 1881 in the village of
2085:Homi Bhabha Centre for Science Education
703:
701:
1280:MacTutor History of Mathematics archive
1088:MacTutor History of Mathematics archive
954:MacTutor History of Mathematics archive
935:
923:
910:
808:
592:
566:
550:
628:https://doi.org/10.1515/opli-2018-0001
1299:, YouTube video, University of Oxford
1285:Ch. 6 ā The Bakhshali manuscript
189:influenced by contemporary dialects.
7:
708:Devlin, Hannah (13 September 2017).
558:
1610:Subbayya Sivasankaranarayana Pillai
2070:Institute of Mathematical Sciences
1094:from the original on 9 August 2007
995:Mason, Robyn (14 September 2017).
93:with influence from local dialects
25:
2080:Harish-Chandra Research Institute
1590:K. R. Parthasarathy (probabilist)
269:, solutions of simple equations,
54:One of the Bakhshali manuscripts.
1308:https://doi.org/10.18732/H2XT07.
1304:History of Science in South Asia
1181:History of Science in South Asia
1171:; Agathe Keller; Takao Hayashi;
1032:History of Science in South Asia
197:The manuscript was unearthed in
48:
2116:1881 archaeological discoveries
1154:. New Delhi: Aditya Prakashan.
2065:Chennai Mathematical Institute
1434:Melpathur Narayana Bhattathiri
977:, New Delhi: Sahitya Akademi,
1:
2126:History of Khyber Pakhtunkhwa
385:(mathematics) chapter of the
271:simultaneous linear equations
2055:Indian Statistical Institute
1150:Kaye, George Rusby (2004) .
856:www.thehindubusinessline.com
828:Bibhutibhusan Datta (1929).
400:approximately, dealing with
2121:Bodleian Library collection
2075:Indian Institute of Science
2060:Bhaskaracharya Pratishthana
1938:A. A. Krishnaswamy Ayyangar
1903:Shankar Balakrishna Dikshit
1830:HinduāArabic numeral system
788:. Oxford University Press.
2152:
1570:Subrahmanyan Chandrasekhar
1230:M N Channabasappa (1976).
1142:On the Bakshali manuscript
1084:"The Bakhshali manuscript"
324:identified as "the son of
1580:Veeravalli S. Varadarajan
1175:; and Dominik Wujastyk. "
117:Too fragile to be handled
47:
39:
27:Ancient mathematical text
1996:Henry Thomas Colebrooke
1585:S. R. Srinivasa Varadhan
1276:The Bakhshali manuscript
303:Buddhist Hybrid Sanskrit
82:, (present-day) Pakistan
2136:Mathematics manuscripts
1469:MÄdhava of Saį¹
gamagrÄma
1117:Hayashi, Takao (1995).
529:Bakhshali approximation
279:indeterminate equations
1067:Joseph, G. G. (2000),
675:"BakhshÄlÄ« Manuscript"
673:Takao Hayashi (2008),
499:
489:
417:
281:of the second degree.
267:geometric progressions
241:
43:, University of Oxford
1680:BrÄhmasphuį¹asiddhÄnta
1239:Indian J. History Sci
834:Bull. Amer. Math. Soc
524:Birch bark manuscript
415:
381:'s commentary on the
239:
1956:T. A. Saraswati Amma
1667:Bakhshali manuscript
1635:Kannan Soundararajan
1007:on 14 September 2017
973:Singh, Maan (1993).
885:Mathematics in India
151:Bakhshali manuscript
35:Bakhshali manuscript
2032:Islamic mathematics
1991:Walter Eugene Clark
1785:Vasishtha Siddhanta
1757:SiddhÄnta Shiromani
1736:Paitamaha Siddhanta
1615:Tilak Raj Prabhakar
1560:Satyendra Nath Bose
1555:Srinivasa Ramanujan
1509:Nilakantha Somayaji
950:"A history of zero"
756:. 14 September 2017
275:quadratic equations
18:Bakshali manuscript
2131:Indian mathematics
1524:Gangesha Upadhyaya
1345:Indian mathematics
754:Bodleian Libraries
534:Indian mathematics
418:
242:
211:Khyber Pakhtunkhwa
179:Indian mathematics
2103:
2102:
1909:Sudhakara Dvivedi
1729:Paulisa Siddhanta
1648:
1647:
1504:Jagannatha Samrat
1499:Achyuta Pisharati
1173:Clemency Montelle
1161:978-81-7742-058-6
1137:Hoernle, Augustus
1128:978-90-6980-087-5
895:978-0-691-12067-6
604:Clemency Montelle
474:Clemency Montelle
468:, Agathe Keller,
447:radiocarbon dated
408:Numerals and zero
209:, in present-day
147:
146:
63:Mathematical text
16:(Redirected from
2143:
1743:Romaka Siddhanta
1625:Akshay Venkatesh
1595:M. S. Narasimhan
1565:P.C. Mahalanobis
1358:
1338:
1331:
1324:
1315:
1265:
1263:
1255:Jonathan Borwein
1246:
1236:
1225:
1224:
1222:
1216:
1209:
1183:, 5.1: 134ā150.
1165:
1146:
1132:
1104:
1103:
1101:
1099:
1079:
1073:
1072:
1064:
1058:
1057:
1047:
1023:
1017:
1016:
1014:
1012:
1003:. Archived from
992:
986:
985:, pp. 9ā11.
971:
965:
964:
962:
960:
945:
939:
933:
927:
920:
914:
908:
899:
898:
876:
867:
866:
864:
862:
848:
842:
841:
825:
812:
806:
800:
799:
779:
766:
765:
763:
761:
746:
733:
732:
730:
728:
705:
696:
695:
670:
639:
636:
630:
623:
610:
600:
580:
576:
570:
555:
487:
459:Saffarid dynasty
457:), and 885ā993 (
402:negative numbers
226:Bodleian Library
215:A. F. R. HoernlƩ
205:, which is near
52:
41:Bodleian Library
32:
21:
2151:
2150:
2146:
2145:
2144:
2142:
2141:
2140:
2106:
2105:
2104:
2099:
2047:
2041:
2005:
1979:
1950:C. T. Rajagopal
1897:Bapudeva Sastri
1889:
1883:
1845:
1825:Brahmi numerals
1817:
1811:
1771:Surya Siddhanta
1644:
1620:Manjul Bhargava
1543:
1402:
1347:
1342:
1306:5 (1). 134ā50.
1272:
1261:
1251:David H. Bailey
1249:
1234:
1229:
1220:
1218:
1217:on 20 June 2014
1214:
1207:
1200:
1197:
1195:Further reading
1189:10.18732/H2XT07
1162:
1149:
1135:
1129:
1116:
1113:
1108:
1107:
1097:
1095:
1081:
1080:
1076:
1066:
1065:
1061:
1045:10.18732/H2XT07
1025:
1024:
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1010:
1008:
994:
993:
989:
972:
968:
958:
956:
947:
946:
942:
934:
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921:
917:
909:
902:
896:
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870:
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858:
850:
849:
845:
827:
826:
815:
807:
803:
796:
781:
780:
769:
759:
757:
748:
747:
736:
726:
724:
707:
706:
699:
693:
672:
671:
642:
637:
633:
624:
613:
601:
594:
589:
584:
583:
577:
573:
556:
552:
547:
520:
488:
485:
443:
410:
366:
287:
234:
195:
169:in present-day
76:Place of origin
55:
28:
23:
22:
15:
12:
11:
5:
2149:
2147:
2139:
2138:
2133:
2128:
2123:
2118:
2108:
2107:
2101:
2100:
2098:
2097:
2092:
2087:
2082:
2077:
2072:
2067:
2062:
2057:
2051:
2049:
2043:
2042:
2040:
2039:
2034:
2029:
2024:
2019:
2013:
2011:
2007:
2006:
2004:
2003:
1998:
1993:
1987:
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1981:
1980:
1978:
1977:
1971:
1965:
1959:
1953:
1947:
1941:
1935:
1930:
1924:
1921:P. C. Sengupta
1918:
1912:
1906:
1900:
1893:
1891:
1885:
1884:
1882:
1881:
1859:
1853:
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1847:
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1838:
1832:
1827:
1821:
1819:
1813:
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1810:
1809:
1802:
1795:
1788:
1781:
1778:Tantrasamgraha
1774:
1767:
1760:
1753:
1746:
1739:
1732:
1725:
1718:
1711:
1704:
1701:Karanapaddhati
1697:
1690:
1687:Ganita Kaumudi
1683:
1676:
1669:
1664:
1656:
1654:
1650:
1649:
1646:
1645:
1643:
1642:
1637:
1632:
1627:
1622:
1617:
1612:
1607:
1605:Harish-Chandra
1602:
1600:C. S. Seshadri
1597:
1592:
1587:
1582:
1577:
1572:
1567:
1562:
1557:
1551:
1549:
1545:
1544:
1542:
1541:
1536:
1534:Sankara Variar
1531:
1526:
1521:
1516:
1511:
1506:
1501:
1496:
1491:
1486:
1481:
1476:
1471:
1466:
1461:
1456:
1451:
1446:
1441:
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1353:Mathematicians
1349:
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1270:External links
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1038:(1): 134ā150.
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286:
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2010:Other regions
2008:
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2001:David Pingree
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1888:Historians of
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1479:Mahendra SÅ«ri
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1245:(2): 112ā124.
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844:
840:(4): 579ā580.
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692:9781402045592
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586:
579:mathematics."
575:
572:
568:
564:
560:
554:
551:
544:
540:
539:Zero (number)
537:
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530:
527:
525:
522:
521:
517:
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511:
509:
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497:
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470:Takao Hayashi
467:
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460:
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451:Indo-Scythian
448:
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361:
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343:
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309:and with Old
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199:British India
192:
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184:
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176:
173:, historical
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38:
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19:
2048:institutions
1968:K. S. Shukla
1940:(1892ā 1953)
1806:Yavanajataka
1804:
1797:
1790:
1783:
1776:
1769:
1764:Åulba SÅ«tras
1762:
1755:
1750:Sadratnamala
1748:
1741:
1734:
1727:
1720:
1713:
1706:
1699:
1694:Kanakkusaram
1692:
1685:
1678:
1671:
1666:
1659:
1529:VarÄhamihira
1494:Parameshvara
1449:GovindasvÄmi
1419:Äryabhaį¹a II
1303:
1242:
1238:
1219:, retrieved
1212:the original
1203:
1180:
1169:Plofker, Kim
1151:
1141:
1118:
1111:Bibliography
1096:. Retrieved
1077:
1068:
1062:
1035:
1031:
1021:
1011:14 September
1009:. Retrieved
1005:the original
1000:
990:
974:
969:
957:. Retrieved
943:
936:Hayashi 1995
931:
924:Hayashi 1995
918:
911:Hayashi 1995
884:
880:Plofker, Kim
859:. Retrieved
855:
846:
837:
833:
809:Hoernle 1887
804:
784:
758:. Retrieved
753:
727:14 September
725:. Retrieved
714:The Guardian
713:
682:
634:
574:
567:Hoernle 1887
553:
512:
500:
495:
494:
490:
479:
463:
453:), 680ā779 (
444:
432:
425:
422:shunya-bindu
421:
419:
398:square roots
391:
382:
374:
370:
367:
355:
349:
346:VarÄhamihira
341:
337:
333:
329:
325:
315:
288:
257:, including
243:
219:
196:
150:
148:
29:
1984:Translators
1976:(1919ā2005)
1974:K. V. Sarma
1970:(1918ā2007)
1964:(1918ā1992)
1958:(1918ā2000)
1952:(1903ā1978)
1946:(1901ā1954)
1944:A. N. Singh
1929:(1888ā1958)
1927:B. B. Datta
1923:(1876ā1962)
1917:(1861ā1916)
1911:(1855ā1910)
1905:(1853ā1898)
1899:(1821ā1900)
1890:mathematics
1834:Symbol for
1818:innovations
1715:Lokavibhaga
1661:Aryabhatiya
1459:Jyeį¹£į¹hadeva
1444:Brahmagupta
1429:BhÄskara II
1414:Äryabhaį¹a I
1398:Yajnavalkya
466:Kim Plofker
455:Turk Shahis
427:Vasavadatta
387:Äryabhaį¹Ä«ya
364:Mathematics
342:MÄrtikÄvata
285:Composition
259:mensuration
87:Language(s)
2110:Categories
1933:T. Hayashi
1816:Pioneering
1799:YuktibhÄį¹£Ä
1722:PÄtÄ«gaį¹ita
1673:Bijaganita
1630:Ravi Vakil
1484:Munishvara
1464:Kamalakara
1439:Brahmadeva
1424:BhÄskara I
1373:Baudhayana
1221:19 January
861:3 February
760:13 January
587:References
508:Bhaskara I
394:algorithms
379:BhÄskara I
307:Apabhraį¹Åa
263:arithmetic
247:arithmetic
222:birch bark
155:birch bark
140:Discovered
135:maths text
101:Birch bark
1962:S. N. Sen
1870:New Delhi
1792:Veį¹vÄroha
1653:Treatises
1454:Halayudha
1407:Classical
1378:Katyayana
1368:Apastamba
1054:2369-775X
722:0261-3077
559:Kaye 2004
504:Aryabhata
351:Takį¹£aÅilÄ
338:rtikÄvati
330:Vasiį¹£į¹ha'
203:Bakhshali
193:Discovery
159:Bakhshali
114:Condition
80:Bakhshali
1878:Varanasi
1836:zero (0)
1708:LÄ«lÄvatÄ«
1575:C.R. Rao
1539:Virasena
1519:Sridhara
1489:Narayana
1474:MahÄvÄ«ra
1257:(2011).
1139:(1887),
1092:Archived
975:Subandhu
882:(2009),
518:See also
484:ā
375:sthÄpanÄ
357:GandhÄra
318:colophon
311:Kashmiri
299:Sanskrit
255:geometry
232:Contents
187:Sanskrit
175:Gandhara
171:Pakistan
167:Peshawar
132:Contents
98:Material
91:Sanskrit
2017:Babylon
1850:Centres
1514:ÅrÄ«pati
1393:Pingala
1361:Ancient
1098:24 July
959:24 July
681:(ed.),
563:Prakrit
435:Gwalior
326:Chajaka
322:brahmin
295:Kashmir
251:algebra
2046:Modern
2037:Europe
2027:Greece
1874:Ujjain
1866:Jaipur
1548:Modern
1388:PÄį¹ini
1383:Manava
1158:
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383:gaį¹ita
334:Hasika
332:s son
207:Mardan
165:(near
163:Mardan
122:Script
106:Format
2022:China
1262:(PDF)
1235:(PDF)
1215:(PDF)
1208:(PDF)
677:, in
545:Notes
371:nyÄsa
2095:TIFR
1223:2016
1156:ISBN
1123:ISBN
1100:2022
1050:ISSN
1013:2017
979:ISBN
961:2022
890:ISBN
863:2022
790:ISBN
762:2023
729:2017
718:ISSN
687:ISBN
441:Date
277:and
265:and
253:and
183:zero
149:The
143:1881
68:Date
60:Type
1185:doi
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369:(
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