402:
39:
226:
485:"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."
302:, 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.
470:
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.
503:
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."
170:". 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
469:
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
490:
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
233:
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
502:
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
614:
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:
366:) 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
491:
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
567:
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
357:
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"
985:
290:, 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
1291:
Plofker, Kim, Agathe Keller, Takao
Hayashi, Clemency Montelle, and Dominik Wujastyk. 2017. "The BakhshÄlÄ« Manuscript: A Response to the Bodleian Libraryās Radiocarbon Datingā.
1926:
1891:
993:
840:
385:
and techniques for a variety of problems, such as systems of linear equations, quadratic equations, arithmetic progressions and arithmetico-geometric series, computing
595:, 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.
2078:
698:
1273:
1080:
738:
1628:
1903:
1191:
409:
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
325:. 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
1324:
2058:
1950:
550:, p. 11), or as the so-called GÄthÄ dialect, the literary form of the Northwestern Prakrit, which combined elements of Sanskrit and
282:, 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
1845:
517:
2020:
2104:
2073:
772:
1268:
1076:
942:
2114:
2083:
2025:
1148:
1115:
882:
60:
AD 224ā383/ 885ā993 (proposed carbon-dates, recently disputed on methodological grounds: Plofker et al. 2017, Houben 2018 Ā§3)
1279:
2109:
465:
and
Dominik Wujastyk to publicly object to the library making the dates globally available, usurping academic precedence:
1598:
2068:
1578:
971:
782:
679:
480:
Referring to the detailed reconsideration of the evidence by
Plofker et al., Sanskrit scholar, Jan Houben remarked:
2124:
1317:
1818:
1239:
1220:
450:). If the dates are valid, it is not known how folios from different centuries came to be collected and buried.
2053:
1668:
1422:
1264:
1072:
1462:
1341:
259:
2043:
1165:
1016:
596:
2119:
2063:
2048:
1310:
872:
206:. After his death, it was examined by G.R.Kaye, who edited the work and published it as a book in 1927.
1558:
475:
Plofker et al., The BakhshÄlÄ« Manuscript: A Response to the
Bodleian Libraryās Radiocarbon Dating, 2017
1015:
Plofker, Kim; Keller, Agathe; Hayashi, Takao; Montelle, Clemency; Wujastyk, Dominik (6 October 2017).
770:
1568:
554:
and whose use as a literary language predated the adoption of Classical Sanskrit for this purposely.(
247:
229:
The numerals used in the Bakhshali manuscript, dated to sometime between the 3rd and 7th century AD.
1858:
1724:
1573:
291:
1247:
1984:
1866:
1457:
986:"Oxford Radiocarbon Accelerator Unit dates the world's oldest recorded origin of the zero symbol"
267:
217:
at the University of Oxford (MS. Sansk. d. 14), though folio are periodically loaned to museums.
1200:
739:"Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'"
663:
2005:
1932:
251:
1745:
422:
Prior to the 2017 carbon dating, a 9th-century inscription of zero on the wall of a temple in
1862:
1854:
1356:
512:
306:
255:
1944:
1623:
1248:"A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics"
413:(literally, the dot of the empty place). References to the concept are found in Subandhu's
375:
819:"Book Review: G. R. Kaye, The BakhshĆ¢lĆ® ManuscriptāA Study in Mediaeval Mathematics, 1927"
8:
2010:
1979:
1787:
1773:
1710:
1603:
1548:
1543:
1497:
1447:
1129:
938:
672:
Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures
263:
143:
203:
1512:
1333:
522:
435:
199:
167:
2015:
1897:
1717:
1492:
1487:
1361:
1161:
1144:
1111:
1038:
967:
878:
778:
771:
John Newsome Crossley; Anthony Wah-Cheung Lun; Kangshen Shen; Shen Kangsheng (1999).
706:
675:
592:
462:
299:
295:
1731:
1613:
1583:
1553:
1243:
1173:
1125:
1028:
1017:"The BakhshÄlÄ« Manuscript: A Response to the Bodleian Library's Radiocarbon Dating"
447:
401:
214:
29:
1696:
1467:
818:
1938:
1885:
1813:
1759:
1608:
1477:
1166:
The BakhshÄlÄ« Manuscript: A Response to the Bodleian Libraryās Radiocarbon Dating
453:
The publication of the radio carbon dates, initially via non-academic media, led
426:, Madhya Pradesh, was once thought to be the oldest Indian use of a zero symbol.
390:
1517:
1437:
334:
286:
and neighbouring regions. The language of the manuscript, though intended to be
1909:
1829:
1766:
1689:
1675:
1593:
1588:
1522:
1366:
627:
All the pages have been photographed which are available in the book by Hayashi
439:
279:
114:
1417:
1285:
378:, including the emphasis on verification that became obsolete in later works.
2098:
1989:
1921:
1850:
1752:
1042:
710:
667:
527:
458:
187:
1412:
1376:
367:
1956:
1794:
1738:
1682:
1502:
1482:
1407:
699:"Much ado about nothing: ancient Indian text contains earliest zero symbol"
1962:
1915:
1703:
1649:
1432:
1386:
1296:
1157:
868:
616:
499:'s work (AD 629), and said that it was "not much later than Bhaskara I".
454:
443:
415:
386:
597:
https://journals.library.ualberta.ca/hssa/index.php/hssa/article/view/22
349:
etc.), the supposed place where the manuscript might have been written.
1661:
1618:
1472:
1452:
1427:
496:
339:
235:
225:
210:
38:
419:, which has been dated between 385 and 465 by the scholar Maan Singh.
1780:
1442:
1402:
1302:
492:
393:(profit and loss), measurement such as of the fineness of gold, etc.
382:
209:
The extant manuscript is incomplete, consisting of seventy leaves of
191:
147:
68:
1286:"A Big Zero: Research uncovers the date of the Bakhshali Manuscript"
1177:
1033:
1563:
1527:
1507:
1134:, Vienna: Alfred Hƶlder (Editor of the Court and of the University)
774:
The Nine Chapters on the Mathematical Art: Companion and Commentary
345:
287:
243:
175:
163:
159:
155:
79:
1193:
The Bakhshali Manuscript: An Ancient Treatise of Indian Arithmetic
1381:
1238:
1108:
The BakhshÄlÄ« manuscript: an ancient Indian mathematical treatise
1070:
936:
551:
423:
310:
283:
239:
1629:
Shanti Swarup Bhatnagar Prize recipients in Mathematical Science
1199:, Allahabad: Dr. Ratna Kumari Svadhyaya Sansthan, archived from
1371:
195:
151:
587:
585:
1299:
Challenges the claims made in the YouTube video "A Big Zero."
495:. Hayashi noted some similarities between the manuscript and
202:, Pakistan. The first research on the manuscript was done by
1014:
1276:(Ian G. Pearce, Indian Mathematics: Redressing the balance)
1058:
The Crest of the Peacock, non-European roots of Mathematics
582:
438:
to three different centuries and empires, from AD 224ā383 (
171:
1830:
Infinite series expansions for the trigonometric functions
1190:
Sarasvati, Svami Satya Prakash; Jyotishmati, Usha (1979),
1141:
The BakhshÄlÄ« manuscripts: a study in medieval mathematics
1110:. Groningen Oriental studies. Groningen: Egbert Forsten.
841:"London museum showcases India's contribution to science"
294:. The overlying dialects, though sharing affinities with
1221:"On the square root formula in the Bakhshali manuscript"
895:
893:
546:
Variously described either as an "irregular Sanskrit" (
309:
to one of the sections states that it was written by a
213:, whose intended order is not known. It is kept at the
250:. The topics covered include fractions, square roots,
2079:
Ramanujan Institute for Advanced Study in Mathematics
1824:
918:
890:
190:
from a field in 1881, by a peasant in the village of
1189:
816:
405:
Bakhshali manuscript, detail of the numeral "zero".
791:
733:
731:
729:
727:
434:In 2017, samples from 3 folios of the corpus were
166:). It is perhaps "the oldest extant manuscript in
142:is an ancient Indian mathematical text written on
1218:
1071:J. J. O'Connor; E. F. Robertson (November 2000).
937:J. J. O'Connor; E. F. Robertson (November 2000).
2096:
278:The manuscript is written in an earlier form of
724:
1060:, Princeton University Press, pp. 215ā216
766:
764:
762:
760:
657:
655:
653:
1318:
930:
812:
810:
808:
806:
661:
651:
649:
647:
645:
643:
641:
639:
637:
635:
633:
591:Plofker, Kim, Agathe Keller, Takao Hayashi,
174:symbol. It is written in a form of literary
1215:with complete text in Devanagari, 110 pages
990:School of Archaeology, University of Oxford
911:Section VII 11, corresponding to folio 46.(
877:, Princeton University Press, p. 158,
757:
337:as being in northwestern India (along with
317:", a "king of calculators," for the use of
1846:Kerala school of astronomy and mathematics
1325:
1311:
1280:Hoernle: On the Bakhshali Manuscript, 1887
863:
861:
803:
630:
329:, which has been interpreted as the place
1032:
674:, vol. 1, Springer, pp. B1āB3,
610:
608:
606:
604:
146:that was found in 1881 in the village of
2074:Homi Bhabha Centre for Science Education
1064:
692:
690:
400:
224:
1269:MacTutor History of Mathematics archive
1124:
1105:
1077:MacTutor History of Mathematics archive
943:MacTutor History of Mathematics archive
924:
912:
899:
867:
858:
797:
555:
2097:
1332:
1055:
956:
696:
617:https://doi.org/10.1515/opli-2018-0001
601:
1306:
1288:, YouTube video, University of Oxford
1274:Ch. 6 ā The Bakhshali manuscript
983:
687:
178:influenced by contemporary dialects.
1138:
697:Devlin, Hannah (13 September 2017).
547:
396:
1599:Subbayya Sivasankaranarayana Pillai
13:
2059:Institute of Mathematical Sciences
1183:
1083:from the original on 9 August 2007
984:Mason, Robyn (14 September 2017).
82:with influence from local dialects
14:
2136:
2069:Harish-Chandra Research Institute
1579:K. R. Parthasarathy (probabilist)
1258:
258:, solutions of simple equations,
43:One of the Bakhshali manuscripts.
1297:https://doi.org/10.18732/H2XT07.
1293:History of Science in South Asia
1170:History of Science in South Asia
1160:; Agathe Keller; Takao Hayashi;
1021:History of Science in South Asia
186:The manuscript was unearthed in
37:
2105:1881 archaeological discoveries
1143:. New Delhi: Aditya Prakashan.
1099:
1049:
1008:
977:
905:
833:
561:
540:
2054:Chennai Mathematical Institute
1423:Melpathur Narayana Bhattathiri
966:, New Delhi: Sahitya Akademi,
621:
352:
273:
1:
2115:History of Khyber Pakhtunkhwa
575:
374:(mathematics) chapter of the
260:simultaneous linear equations
2044:Indian Statistical Institute
1139:Kaye, George Rusby (2004) .
845:www.thehindubusinessline.com
817:Bibhutibhusan Datta (1929).
389:approximately, dealing with
181:
7:
2110:Bodleian Library collection
2064:Indian Institute of Science
2049:Bhaskaracharya Pratishthana
1927:A. A. Krishnaswamy Ayyangar
1892:Shankar Balakrishna Dikshit
1819:HinduāArabic numeral system
777:. Oxford University Press.
506:
220:
10:
2141:
1559:Subrahmanyan Chandrasekhar
1219:M N Channabasappa (1976).
1131:On the Bakshali manuscript
1073:"The Bakhshali manuscript"
313:identified as "the son of
2034:
1998:
1972:
1876:
1838:
1804:
1641:
1569:Veeravalli S. Varadarajan
1536:
1395:
1349:
1340:
1164:; and Dominik Wujastyk. "
128:
120:
110:
106:Too fragile to be handled
102:
94:
86:
75:
64:
56:
48:
36:
28:
23:
16:Ancient mathematical text
1985:Henry Thomas Colebrooke
1574:S. R. Srinivasa Varadhan
1265:The Bakhshali manuscript
533:
292:Buddhist Hybrid Sanskrit
71:, (present-day) Pakistan
2125:Mathematics manuscripts
1458:MÄdhava of Saį¹
gamagrÄma
1106:Hayashi, Takao (1995).
518:Bakhshali approximation
429:
268:indeterminate equations
1056:Joseph, G. G. (2000),
664:"BakhshÄlÄ« Manuscript"
662:Takao Hayashi (2008),
488:
478:
406:
270:of the second degree.
256:geometric progressions
230:
32:, University of Oxford
1669:BrÄhmasphuį¹asiddhÄnta
1228:Indian J. History Sci
823:Bull. Amer. Math. Soc
513:Birch bark manuscript
482:
467:
404:
370:'s commentary on the
228:
1945:T. A. Saraswati Amma
1656:Bakhshali manuscript
1624:Kannan Soundararajan
996:on 14 September 2017
962:Singh, Maan (1993).
874:Mathematics in India
140:Bakhshali manuscript
24:Bakhshali manuscript
2021:Islamic mathematics
1980:Walter Eugene Clark
1774:Vasishtha Siddhanta
1746:SiddhÄnta Shiromani
1725:Paitamaha Siddhanta
1604:Tilak Raj Prabhakar
1549:Satyendra Nath Bose
1544:Srinivasa Ramanujan
1498:Nilakantha Somayaji
939:"A history of zero"
745:. 14 September 2017
264:quadratic equations
2120:Indian mathematics
1513:Gangesha Upadhyaya
1334:Indian mathematics
743:Bodleian Libraries
523:Indian mathematics
407:
231:
200:Khyber Pakhtunkhwa
168:Indian mathematics
2092:
2091:
1898:Sudhakara Dvivedi
1718:Paulisa Siddhanta
1637:
1636:
1493:Jagannatha Samrat
1488:Achyuta Pisharati
1162:Clemency Montelle
1150:978-81-7742-058-6
1126:Hoernle, Augustus
1117:978-90-6980-087-5
884:978-0-691-12067-6
593:Clemency Montelle
463:Clemency Montelle
457:, Agathe Keller,
436:radiocarbon dated
397:Numerals and zero
198:, in present-day
136:
135:
52:Mathematical text
2132:
1732:Romaka Siddhanta
1614:Akshay Venkatesh
1584:M. S. Narasimhan
1554:P.C. Mahalanobis
1347:
1346:
1327:
1320:
1313:
1304:
1303:
1254:
1252:
1244:Jonathan Borwein
1235:
1225:
1214:
1213:
1211:
1205:
1198:
1172:, 5.1: 134ā150.
1154:
1135:
1121:
1093:
1092:
1090:
1088:
1068:
1062:
1061:
1053:
1047:
1046:
1036:
1012:
1006:
1005:
1003:
1001:
992:. Archived from
981:
975:
974:, pp. 9ā11.
960:
954:
953:
951:
949:
934:
928:
922:
916:
909:
903:
897:
888:
887:
865:
856:
855:
853:
851:
837:
831:
830:
814:
801:
795:
789:
788:
768:
755:
754:
752:
750:
735:
722:
721:
719:
717:
694:
685:
684:
659:
628:
625:
619:
612:
599:
589:
569:
565:
559:
544:
476:
448:Saffarid dynasty
446:), and 885ā993 (
391:negative numbers
215:Bodleian Library
204:A. F. R. HoernlƩ
194:, which is near
41:
30:Bodleian Library
21:
20:
2140:
2139:
2135:
2134:
2133:
2131:
2130:
2129:
2095:
2094:
2093:
2088:
2036:
2030:
1994:
1968:
1939:C. T. Rajagopal
1886:Bapudeva Sastri
1878:
1872:
1834:
1814:Brahmi numerals
1806:
1800:
1760:Surya Siddhanta
1633:
1609:Manjul Bhargava
1532:
1391:
1336:
1331:
1295:5 (1). 134ā50.
1261:
1250:
1240:David H. Bailey
1223:
1209:
1207:
1206:on 20 June 2014
1203:
1196:
1186:
1184:Further reading
1178:10.18732/H2XT07
1151:
1118:
1102:
1097:
1096:
1086:
1084:
1069:
1065:
1054:
1050:
1034:10.18732/H2XT07
1013:
1009:
999:
997:
982:
978:
961:
957:
947:
945:
935:
931:
923:
919:
910:
906:
898:
891:
885:
866:
859:
849:
847:
839:
838:
834:
815:
804:
796:
792:
785:
769:
758:
748:
746:
737:
736:
725:
715:
713:
695:
688:
682:
660:
631:
626:
622:
613:
602:
590:
583:
578:
573:
572:
566:
562:
545:
541:
536:
509:
477:
474:
432:
399:
355:
276:
223:
184:
158:in present-day
65:Place of origin
44:
17:
12:
11:
5:
2138:
2128:
2127:
2122:
2117:
2112:
2107:
2090:
2089:
2087:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2040:
2038:
2032:
2031:
2029:
2028:
2023:
2018:
2013:
2008:
2002:
2000:
1996:
1995:
1993:
1992:
1987:
1982:
1976:
1974:
1970:
1969:
1967:
1966:
1960:
1954:
1948:
1942:
1936:
1930:
1924:
1919:
1913:
1910:P. C. Sengupta
1907:
1901:
1895:
1889:
1882:
1880:
1874:
1873:
1871:
1870:
1848:
1842:
1840:
1836:
1835:
1833:
1832:
1827:
1821:
1816:
1810:
1808:
1802:
1801:
1799:
1798:
1791:
1784:
1777:
1770:
1767:Tantrasamgraha
1763:
1756:
1749:
1742:
1735:
1728:
1721:
1714:
1707:
1700:
1693:
1690:Karanapaddhati
1686:
1679:
1676:Ganita Kaumudi
1672:
1665:
1658:
1653:
1645:
1643:
1639:
1638:
1635:
1634:
1632:
1631:
1626:
1621:
1616:
1611:
1606:
1601:
1596:
1594:Harish-Chandra
1591:
1589:C. S. Seshadri
1586:
1581:
1576:
1571:
1566:
1561:
1556:
1551:
1546:
1540:
1538:
1534:
1533:
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2037:institutions
1957:K. S. Shukla
1929:(1892ā 1953)
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1753:Åulba SÅ«tras
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1683:Kanakkusaram
1681:
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1483:Parameshvara
1438:GovindasvÄmi
1408:Äryabhaį¹a II
1292:
1231:
1227:
1208:, retrieved
1201:the original
1192:
1169:
1158:Plofker, Kim
1140:
1130:
1107:
1100:Bibliography
1085:. Retrieved
1066:
1057:
1051:
1024:
1020:
1010:
1000:14 September
998:. Retrieved
994:the original
989:
979:
963:
958:
946:. Retrieved
932:
925:Hayashi 1995
920:
913:Hayashi 1995
907:
900:Hayashi 1995
873:
869:Plofker, Kim
848:. Retrieved
844:
835:
826:
822:
798:Hoernle 1887
793:
773:
747:. Retrieved
742:
716:14 September
714:. Retrieved
703:The Guardian
702:
671:
623:
563:
556:Hoernle 1887
542:
501:
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484:
483:
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468:
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442:), 680ā779 (
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411:shunya-bindu
410:
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387:square roots
380:
371:
363:
359:
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344:
338:
335:VarÄhamihira
330:
326:
322:
318:
314:
304:
277:
246:, including
232:
208:
185:
139:
137:
18:
1973:Translators
1965:(1919ā2005)
1963:K. V. Sarma
1959:(1918ā2007)
1953:(1918ā1992)
1947:(1918ā2000)
1941:(1903ā1978)
1935:(1901ā1954)
1933:A. N. Singh
1918:(1888ā1958)
1916:B. B. Datta
1912:(1876ā1962)
1906:(1861ā1916)
1900:(1855ā1910)
1894:(1853ā1898)
1888:(1821ā1900)
1879:mathematics
1823:Symbol for
1807:innovations
1704:Lokavibhaga
1650:Aryabhatiya
1448:Jyeį¹£į¹hadeva
1433:Brahmagupta
1418:BhÄskara II
1403:Äryabhaį¹a I
1387:Yajnavalkya
455:Kim Plofker
444:Turk Shahis
416:Vasavadatta
376:Äryabhaį¹Ä«ya
353:Mathematics
331:MÄrtikÄvata
274:Composition
248:mensuration
76:Language(s)
2099:Categories
1922:T. Hayashi
1805:Pioneering
1788:YuktibhÄį¹£Ä
1711:PÄtÄ«gaį¹ita
1662:Bijaganita
1619:Ravi Vakil
1473:Munishvara
1453:Kamalakara
1428:Brahmadeva
1413:BhÄskara I
1362:Baudhayana
1210:19 January
850:3 February
749:13 January
576:References
497:Bhaskara I
383:algorithms
368:BhÄskara I
296:Apabhraį¹Åa
252:arithmetic
236:arithmetic
211:birch bark
144:birch bark
129:Discovered
124:maths text
90:Birch bark
1951:S. N. Sen
1859:New Delhi
1781:Veį¹vÄroha
1642:Treatises
1443:Halayudha
1396:Classical
1367:Katyayana
1357:Apastamba
1043:2369-775X
711:0261-3077
548:Kaye 2004
493:Aryabhata
340:Takį¹£aÅilÄ
327:rtikÄvati
319:Vasiį¹£į¹ha'
192:Bakhshali
182:Discovery
148:Bakhshali
103:Condition
69:Bakhshali
1867:Varanasi
1825:zero (0)
1697:LÄ«lÄvatÄ«
1564:C.R. Rao
1528:Virasena
1508:Sridhara
1478:Narayana
1463:MahÄvÄ«ra
1246:(2011).
1128:(1887),
1081:Archived
964:Subandhu
871:(2009),
507:See also
473:ā
364:sthÄpanÄ
346:GandhÄra
307:colophon
300:Kashmiri
288:Sanskrit
244:geometry
221:Contents
176:Sanskrit
164:Gandhara
160:Pakistan
156:Peshawar
121:Contents
87:Material
80:Sanskrit
2006:Babylon
1839:Centres
1503:ÅrÄ«pati
1382:Pingala
1350:Ancient
1087:24 July
948:24 July
670:(ed.),
552:Prakrit
424:Gwalior
315:Chajaka
311:brahmin
284:Kashmir
240:algebra
2035:Modern
2026:Europe
2016:Greece
1863:Ujjain
1855:Jaipur
1537:Modern
1377:PÄį¹ini
1372:Manava
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372:gaį¹ita
323:Hasika
321:s son
196:Mardan
154:(near
152:Mardan
111:Script
95:Format
2011:China
1251:(PDF)
1224:(PDF)
1204:(PDF)
1197:(PDF)
666:, in
534:Notes
360:nyÄsa
2084:TIFR
1212:2016
1145:ISBN
1112:ISBN
1089:2022
1039:ISSN
1002:2017
968:ISBN
950:2022
879:ISBN
852:2022
779:ISBN
751:2023
718:2017
707:ISSN
676:ISBN
430:Date
266:and
254:and
242:and
172:zero
138:The
132:1881
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49:Type
1174:doi
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
