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Bakhshali manuscript

February 16, 2023

The Bakhshali manuscript is an ancient mathematical text written on birch bark that was found in 1881 in the village of Bakhshali, Mardan (near Peshawar in present-day Pakistan, historical Gandhara). It is perhaps "the oldest extant manuscript in Indian mathematics."[4] For some portions a carbon-date was proposed of AD 224–383 while for other portions a carbon-date as late as AD 885–993 in a recent study, but the dating has been criticised by specialists on methodological grounds (Plofker et al. 2017[1] and Houben 2018 §3[2]). The manuscript contains the earliest known Indian use of a zero symbol.[5][6] It is written in a form of literary Sanskrit influenced by contemporary dialects.

Bakhshali manuscript
Bodleian Library, University of Oxford
One of the Bakhshali manuscripts.
TypeMathematical text
DateAD 224–383/ 885–993 (proposed carbon-dates, recently disputed on methodological grounds: Plofker et al. 2017,[1] Houben 2018 §3[2])
Place of originBakhshali, Pakistan
Language(s)Sanskrit with influence from local dialects
MaterialBirch bark
FormatSeventy leaves
ConditionToo fragile to be handled[3]
ScriptSharada script
Contentsmaths text
Discovered1881

Discovery

The manuscript was unearthed from a field in 1881,[7] by a peasant in the village of Bakhshali, which is near Mardan, in Khyber Pakhtunkhwa, Pakistan.[4] The first research on the manuscript was done by A. F. R. Hoernlé.[4][8] After his death, it was examined by G. R. Kaye, who edited the work and published it as a book in 1927.[9]

The extant manuscript is incomplete, consisting of seventy leaves of birch bark,[4][7] whose intended order is not known.[4] It is kept at the Bodleian Library at the University of Oxford[4][7] (MS. Sansk. d. 14), though folio are periodically loaned to museums.[10]

Contents

 
The numerals used in the Bakhshali manuscript, dated to sometime between the 3rd and 7th century AD.

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 arithmetic, algebra and geometry, including mensuration. The topics covered include fractions, square roots, arithmetic and geometric progressions, solutions of simple equations, simultaneous linear equations, quadratic equations and indeterminate equations of the second degree.[9][11]

Composition

The manuscript is written in an earlier form of Sharada script, 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 Kashmir and neighbouring regions.[4] The language of the manuscript,[a] though intended to be Sanskrit, was significantly influenced in its phonetics and morphology by a local dialect or dialects, and some of the resultant linguistic peculiarities of the text are shared with Buddhist Hybrid Sanskrit. The overlying dialects, though sharing affinities with Apabhraṃśa and with Old Kashmiri, have not been identified precisely.[12] 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.[13] It is possible that the manuscript might be a compilation of fragments from different works composed in a number of language varieties.[12] Hayashi admits that some of the irregularities are due to errors by scribes or may be orthographical.[14]

A colophon to one of the sections states that it was written by a brahmin identified as "the son of Chajaka", a "king of calculators," for the use of Vasiṣṭha's son Hasika. The brahmin might have been the author of the commentary as well as the scribe of the manuscript.[11] Near the colophon appears a broken word rtikāvati, which has been interpreted as the place Mārtikāvata mentioned by Varāhamihira as being in northwestern India (along with Takṣaśilā, Gandhāra etc.), the supposed place where the manuscript might have been written.[4]

Mathematics

The manuscript is a compilation of mathematical rules and examples (in verse), and prose commentaries on these verses.[4] Typically, a rule is given, with one or more examples, where each example is followed by a "statement" (nyāsa / sthāpanā) 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.[4] This is a style similar to that of Bhāskara I's commentary on the gaṇita (mathematics) chapter of the Āryabhaṭīya, including the emphasis on verification that became obsolete in later works.[4]

The rules are algorithms and techniques for a variety of problems, such as systems of linear equations, quadratic equations, arithmetic progressions and arithmetico-geometric series, computing square roots approximately, dealing with negative numbers (profit and loss), measurement such as of the fineness of gold, etc.[7]

Mathematical context

Scholar Takao Hayashi has compared the text of the manuscript with several Sanskrit texts.[4] He mentions that a passage is a verbatim quote from Mahabharata. He discusses similar passages in Ramayana, Vayupurana, Lokaprakasha of Kshemendra etc. Some of the mathematical rules also appear in Aryabhatiya of Aryabhatta, Aryabhatiyabhashya of Bhaskara I, Patiganita and Trairashika of Sridhara, Ganitasarasamgraha of Mahavira, and Lilavati and Bijaganita of Bhaskara II. An unnamed manuscript, later than Thakkar Pheru, in the Patan Jain library, a compilation of mathematical rules from various sources resembles the Bakhshali manuscript, contains data in an example which are strikingly similar.[citation needed]

Numerals and zero

 
Bakhshali manuscript, detail of the numeral "zero".

The Bakhshali manuscript uses numerals with a place-value system, using a dot as a place holder for zero.[15][6] The dot symbol came to be called the shunya-bindu (literally, the dot of the empty place). References to the concept are found in Subandhu's Vasavadatta, which has been dated between 385 and 465 by the scholar Maan Singh.[16]

Prior to the 2017 carbon dating a 9th-century inscription of zero on the wall of a temple in Gwalior, Madhya Pradesh, was once thought to be the oldest Indian use of a zero symbol.[6]

Date

In 2017, samples from 3 folios of the corpus were radiocarbon dated to three different centuries and empires, from AD 224–383 (Indo-Scythian), 680–779 (Turk Shahis), and 885–993 (Saffarid dynasty). If the dates are valid, it is not known how folios from different centuries came to collected, and buried.[5][17][6]

The publication of the radio carbon dates, initially via non academic media, led Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk to publicly object to the library making the dates globally available, usurping academic precedence:

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 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.

— Plofker et al., The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating, 2017[18]

Referring to the detailed reconsideration of the evidence by Plofker et al., Sanskrit scholar, Jan Houben remarked:

"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."[2]

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. Hoernlé thought that the manuscript was from the 9th century, but the original was from the 3rd or 4th century.[b] Indian scholars assigned it an earlier date. Datta assigned it to the "early centuries of the Christian era".[9] Channabasappa dated it to AD 200–400, on the grounds that it uses mathematical terminology different from that of Aryabhata.[20] Hayashi noted some similarities between the manuscript and Bhaskara I's work (AD 629), and said that it was "not much later than Bhaskara I".[4]

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 necessary to "take quite seriously the judgement of palaeographists such as Richard Salomon who observed that, what he teleologically called “Proto-Śāradā,” “first emerged around the middle of the seventh century” (Salomon 1998: 40). This excludes the earlier dates attributed to manuscript folios on which a fully developed form of Śāradā appears."[2]

See also

Notes

  1. ^ Variously described either as an "irregular Sanskrit" (Kaye 2004, p. 11), or as the so-called Gāthā dialect, the literary form of the Northwestern Prakrit, which combined elements of Sanskrit and Prakrit and whose use as a literary language predated the adoption of Classical Sanskrit for this purpose.(Hoernle 1887, p. 10)
  2. ^ G. R. Kaye, on the other hand, thought in 1927 that the work was composed in the 12th century,[4][9] 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 mathematics."[19]

References

  1. ^ a b Plofker, Kim, Agathe Keller, Takao Hayashi, Clemency Montelle, and Dominik Wujastyk. 2017. “The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating.” History of Science in South Asia, 5.1: 134-150. https://journals.library.ualberta.ca/hssa/index.php/hssa/article/view/22
  2. ^ a b c d 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 – Vol. 4, issue 1: 1-18. DOI: https://doi.org/10.1515/opli-2018-0001
  3. ^ All the pages have been photographed which are available in the book by Hayashi
  4. ^ a b c d e f g h i j k l m n Takao Hayashi (2008), "Bakhshālī Manuscript", in Helaine Selin (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, vol. 1, Springer, pp. B1–B3, ISBN 9781402045592
  5. ^ a b Devlin, Hannah (13 September 2017). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Retrieved 14 September 2017.
  6. ^ a b c d "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Libraries. 14 September 2017. Retrieved 13 January 2023.
  7. ^ a b c d John Newsome Crossley; Anthony Wah-Cheung Lun; Kangshen Shen; Shen Kangsheng (1999). The Nine Chapters on the Mathematical Art: Companion and Commentary. Oxford University Press. ISBN 0-19-853936-3.
  8. ^ Hoernle 1887.
  9. ^ a b c d Bibhutibhusan Datta (1929). "Book Review: G. R. Kaye, The Bakhshâlî Manuscript—A Study in Mediaeval Mathematics, 1927". 35 (4). Bull. Amer. Math. Soc.: 579–580. {{cite journal}}: Cite journal requires |journal= (help)
  10. ^ "London museum showcases India's contribution to science". www.thehindubusinessline.com. Retrieved 3 February 2022.
  11. ^ a b Plofker, Kim (2009), Mathematics in India, Princeton University Press, p. 158, ISBN 978-0-691-12067-6
  12. ^ a b Hayashi 1995, p. 54.
  13. ^ Section VII 11, corresponding to folio 46v.(Hayashi 1995, p. 54)
  14. ^ Hayashi 1995, p. 26.
  15. ^ J. J. O'Connor; E. F. Robertson (November 2000). "A history of zero". MacTutor History of Mathematics archive. Retrieved 24 July 2022.{{cite web}}: CS1 maint: url-status (link)
  16. ^ Singh, Maan (1993). Subandhu, New Delhi: Sahitya Akademi, ISBN 81-7201-509-7, pp. 9–11.
  17. ^ Mason, Robyn (14 September 2017). . School of Archaeology, University of Oxford. Archived from the original on 14 September 2017. Retrieved 14 September 2017.
  18. ^ Plofker, Kim; Keller, Agathe; Hayashi, Takao; Montelle, Clemency; Wujastyk, Dominik (6 October 2017). "The Bakhshālī Manuscript: A Response to the Bodleian Library's Radiocarbon Dating". History of Science in South Asia. 5 (1): 134–150. doi:10.18732/H2XT07. ISSN 2369-775X.
  19. ^ Joseph, G. G. (2000), The Crest of the Peacock, non-European roots of Mathematics, Princeton University Press, pp. 215–216
  20. ^ J. J. O'Connor; E. F. Robertson (November 2000). "The Bakhshali manuscript". MacTutor History of Mathematics archive. from the original on 9 August 2007. Retrieved 24 July 2022.

Bibliography

  • Hayashi, Takao (1995). The Bakhshālī manuscript: an ancient Indian mathematical treatise. Groningen Oriental studies. Groningen: Egbert Forsten. ISBN 978-90-6980-087-5.
  • Hoernle, Augustus (1887), On the Bakshali manuscript, Vienna: Alfred Hölder (Editor of the Court and of the University)
  • Kaye, George Rusby (2004) [1927]. The Bakhshālī manuscripts: a study in medieval mathematics. New Delhi: Aditya Prakashan. ISBN 978-81-7742-058-6.
  • Plofker, Kim; Agathe Keller; Takao Hayashi; Clemency Montelle; and Dominik Wujastyk. "The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating" History of Science in South Asia, 5.1: 134-150. doi:10.18732/H2XT07

Further reading

  • Sarasvati, Svami Satya Prakash; Jyotishmati, Usha (1979), (PDF), Allahabad: Dr. Ratna Kumari Svadhyaya Sansthan, archived from the original (PDF) on 20 June 2014, retrieved 19 January 2016 with complete text in Devanagari, 110 pages
  • M N Channabasappa (1976). "On the square root formula in the Bakhshali manuscript" (PDF). Indian J. History Sci. 11 (2): 112–124.
  • David H. Bailey, Jonathan Borwein (2011). "A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics" (PDF).

External links

  • The Bakhshali manuscript, MacTutor History of Mathematics archive
  • (Ian G. Pearce, Indian Mathematics: Redressing the balance)
  • Hoernle: On the Bakhshali Manuscript, 1887, archive.org
  • "A Big Zero: Research uncovers the date of the Bakhshali Manuscript", YouTube video, University of Oxford
  • Plofker, Kim, Agathe Keller, Takao Hayashi, Clemency Montelle, and Dominik Wujastyk. 2017. “The Bakhshālī Manuscript: A Response to the Bodleian Library’s Radiocarbon Dating”. History of Science in South Asia 5 (1). 134-50. https://doi.org/10.18732/H2XT07. Challenges the claims made in the YouTube video "A Big Zero."

February 16, 2023
bakhshali, manuscript, ancient, mathematical, text, written, birch, bark, that, found, 1881, village, bakhshali, mardan, near, peshawar, present, pakistan, historical, gandhara, perhaps, oldest, extant, manuscript, indian, mathematics, some, portions, carbon, . The Bakhshali manuscript is an ancient mathematical text written on birch bark that was found in 1881 in the village of Bakhshali Mardan near Peshawar in present day Pakistan historical Gandhara It is perhaps the oldest extant manuscript in Indian mathematics 4 For some portions a carbon date was proposed of AD 224 383 while for other portions a carbon date as late as AD 885 993 in a recent study but the dating has been criticised by specialists on methodological grounds Plofker et al 2017 1 and Houben 2018 3 2 The manuscript contains the earliest known Indian use of a zero symbol 5 6 It is written in a form of literary Sanskrit influenced by contemporary dialects Bakhshali manuscriptBodleian Library University of OxfordOne of the Bakhshali manuscripts TypeMathematical textDateAD 224 383 885 993 proposed carbon dates recently disputed on methodological grounds Plofker et al 2017 1 Houben 2018 3 2 Place of originBakhshali PakistanLanguage s Sanskrit with influence from local dialectsMaterialBirch barkFormatSeventy leavesConditionToo fragile to be handled 3 ScriptSharada scriptContentsmaths textDiscovered1881 Contents 1 Discovery 2 Contents 2 1 Composition 2 2 Mathematics 2 2 1 Mathematical context 2 3 Numerals and zero 3 Date 4 See also 5 Notes 6 References 7 Bibliography 8 Further reading 9 External linksDiscovery EditThe manuscript was unearthed from a field in 1881 7 by a peasant in the village of Bakhshali which is near Mardan in Khyber Pakhtunkhwa Pakistan 4 The first research on the manuscript was done by A F R Hoernle 4 8 After his death it was examined by G R Kaye who edited the work and published it as a book in 1927 9 The extant manuscript is incomplete consisting of seventy leaves of birch bark 4 7 whose intended order is not known 4 It is kept at the Bodleian Library at the University of Oxford 4 7 MS Sansk d 14 though folio are periodically loaned to museums 10 Contents Edit The numerals used in the Bakhshali manuscript dated to sometime between the 3rd and 7th century AD 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 arithmetic algebra and geometry including mensuration The topics covered include fractions square roots arithmetic and geometric progressions solutions of simple equations simultaneous linear equations quadratic equations and indeterminate equations of the second degree 9 11 Composition Edit The manuscript is written in an earlier form of Sharada script 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 Kashmir and neighbouring regions 4 The language of the manuscript a though intended to be Sanskrit was significantly influenced in its phonetics and morphology by a local dialect or dialects and some of the resultant linguistic peculiarities of the text are shared with Buddhist Hybrid Sanskrit The overlying dialects though sharing affinities with Apabhraṃsa and with Old Kashmiri have not been identified precisely 12 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 13 It is possible that the manuscript might be a compilation of fragments from different works composed in a number of language varieties 12 Hayashi admits that some of the irregularities are due to errors by scribes or may be orthographical 14 A colophon to one of the sections states that it was written by a brahmin identified as the son of Chajaka a king of calculators for the use of Vasiṣṭha s son Hasika The brahmin might have been the author of the commentary as well as the scribe of the manuscript 11 Near the colophon appears a broken word rtikavati which has been interpreted as the place Martikavata mentioned by Varahamihira as being in northwestern India along with Takṣasila Gandhara etc the supposed place where the manuscript might have been written 4 Mathematics Edit The manuscript is a compilation of mathematical rules and examples in verse and prose commentaries on these verses 4 Typically a rule is given with one or more examples where each example is followed by a statement nyasa sthapana 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 4 This is a style similar to that of Bhaskara I s commentary on the gaṇita mathematics chapter of the Aryabhaṭiya including the emphasis on verification that became obsolete in later works 4 The rules are algorithms and techniques for a variety of problems such as systems of linear equations quadratic equations arithmetic progressions and arithmetico geometric series computing square roots approximately dealing with negative numbers profit and loss measurement such as of the fineness of gold etc 7 Mathematical context Edit Scholar Takao Hayashi has compared the text of the manuscript with several Sanskrit texts 4 He mentions that a passage is a verbatim quote from Mahabharata He discusses similar passages in Ramayana Vayupurana Lokaprakasha of Kshemendra etc Some of the mathematical rules also appear in Aryabhatiya of Aryabhatta Aryabhatiyabhashya of Bhaskara I Patiganita and Trairashika of Sridhara Ganitasarasamgraha of Mahavira and Lilavati and Bijaganita of Bhaskara II An unnamed manuscript later than Thakkar Pheru in the Patan Jain library a compilation of mathematical rules from various sources resembles the Bakhshali manuscript contains data in an example which are strikingly similar citation needed Numerals and zero Edit Bakhshali manuscript detail of the numeral zero The Bakhshali manuscript uses numerals with a place value system using a dot as a place holder for zero 15 6 The dot symbol came to be called the shunya bindu literally the dot of the empty place References to the concept are found in Subandhu s Vasavadatta which has been dated between 385 and 465 by the scholar Maan Singh 16 Prior to the 2017 carbon dating a 9th century inscription of zero on the wall of a temple in Gwalior Madhya Pradesh was once thought to be the oldest Indian use of a zero symbol 6 Date EditIn 2017 samples from 3 folios of the corpus were radiocarbon dated to three different centuries and empires from AD 224 383 Indo Scythian 680 779 Turk Shahis and 885 993 Saffarid dynasty If the dates are valid it is not known how folios from different centuries came to collected and buried 5 17 6 The publication of the radio carbon dates initially via non academic media led Kim Plofker Agathe Keller Takao Hayashi Clemency Montelle and Dominik Wujastyk to publicly object to the library making the dates globally available usurping academic precedence 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 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 Plofker et al The Bakhshali Manuscript A Response to the Bodleian Library s Radiocarbon Dating 2017 18 Referring to the detailed reconsideration of the evidence by Plofker et al Sanskrit scholar Jan Houben remarked 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 2 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 Hoernle thought that the manuscript was from the 9th century but the original was from the 3rd or 4th century b Indian scholars assigned it an earlier date Datta assigned it to the early centuries of the Christian era 9 Channabasappa dated it to AD 200 400 on the grounds that it uses mathematical terminology different from that of Aryabhata 20 Hayashi noted some similarities between the manuscript and Bhaskara I s work AD 629 and said that it was not much later than Bhaskara I 4 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 necessary to take quite seriously the judgement of palaeographists such as Richard Salomon who observed that what he teleologically called Proto Sarada first emerged around the middle of the seventh century Salomon 1998 40 This excludes the earlier dates attributed to manuscript folios on which a fully developed form of Sarada appears 2 See also EditBirch bark manuscript Bakhshali approximation Indian mathematics Zero number Notes Edit Variously described either as an irregular Sanskrit Kaye 2004 p 11 or as the so called Gatha dialect the literary form of the Northwestern Prakrit which combined elements of Sanskrit and Prakrit and whose use as a literary language predated the adoption of Classical Sanskrit for this purpose Hoernle 1887 p 10 G R Kaye on the other hand thought in 1927 that the work was composed in the 12th century 4 9 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 mathematics 19 References Edit a b Plofker Kim Agathe Keller Takao Hayashi Clemency Montelle and Dominik Wujastyk 2017 The Bakhshali Manuscript A Response to the Bodleian Library s Radiocarbon Dating History of Science in South Asia 5 1 134 150 https journals library ualberta ca hssa index php hssa article view 22 a b c d 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 Vol 4 issue 1 1 18 DOI https doi org 10 1515 opli 2018 0001 All the pages have been photographed which are available in the book by Hayashi a b c d e f g h i j k l m n Takao Hayashi 2008 Bakhshali Manuscript in Helaine Selin ed Encyclopaedia of the History of Science Technology and Medicine in Non Western Cultures vol 1 Springer pp B1 B3 ISBN 9781402045592 a b Devlin Hannah 13 September 2017 Much ado about nothing ancient Indian text contains earliest zero symbol The Guardian ISSN 0261 3077 Retrieved 14 September 2017 a b c d Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol zero Bodleian Libraries 14 September 2017 Retrieved 13 January 2023 a b c d John Newsome Crossley Anthony Wah Cheung Lun Kangshen Shen Shen Kangsheng 1999 The Nine Chapters on the Mathematical Art Companion and Commentary Oxford University Press ISBN 0 19 853936 3 Hoernle 1887 a b c d Bibhutibhusan Datta 1929 Book Review G R Kaye The Bakhshali Manuscript A Study in Mediaeval Mathematics 1927 35 4 Bull Amer Math Soc 579 580 a href Template Cite journal html title Template Cite journal cite journal a Cite journal requires journal help London museum showcases India s contribution to science www thehindubusinessline com Retrieved 3 February 2022 a b Plofker Kim 2009 Mathematics in India Princeton University Press p 158 ISBN 978 0 691 12067 6 a b Hayashi 1995 p 54 Section VII 11 corresponding to folio 46v Hayashi 1995 p 54 Hayashi 1995 p 26 J J O Connor E F Robertson November 2000 A history of zero MacTutor History of Mathematics archive Retrieved 24 July 2022 a href Template Cite web html title Template Cite web cite web a CS1 maint url status link Singh Maan 1993 Subandhu New Delhi Sahitya Akademi ISBN 81 7201 509 7 pp 9 11 Mason Robyn 14 September 2017 Oxford Radiocarbon Accelerator Unit dates the world s oldest recorded origin of the zero symbol School of Archaeology University of Oxford Archived from the original on 14 September 2017 Retrieved 14 September 2017 Plofker Kim Keller Agathe Hayashi Takao Montelle Clemency Wujastyk Dominik 6 October 2017 The Bakhshali Manuscript A Response to the Bodleian Library s Radiocarbon Dating History of Science in South Asia 5 1 134 150 doi 10 18732 H2XT07 ISSN 2369 775X Joseph G G 2000 The Crest of the Peacock non European roots of Mathematics Princeton University Press pp 215 216 J J O Connor E F Robertson November 2000 The Bakhshali manuscript MacTutor History of Mathematics archive Archived from the original on 9 August 2007 Retrieved 24 July 2022 Bibliography EditHayashi Takao 1995 The Bakhshali manuscript an ancient Indian mathematical treatise Groningen Oriental studies Groningen Egbert Forsten ISBN 978 90 6980 087 5 Hoernle Augustus 1887 On the Bakshali manuscript Vienna Alfred Holder Editor of the Court and of the University Kaye George Rusby 2004 1927 The Bakhshali manuscripts a study in medieval mathematics New Delhi Aditya Prakashan ISBN 978 81 7742 058 6 Plofker Kim Agathe Keller Takao Hayashi Clemency Montelle and Dominik Wujastyk The Bakhshali Manuscript A Response to the Bodleian Library s Radiocarbon Dating History of Science in South Asia 5 1 134 150 doi 10 18732 H2XT07Further reading EditSarasvati Svami Satya Prakash Jyotishmati Usha 1979 The Bakhshali Manuscript An Ancient Treatise of Indian Arithmetic PDF Allahabad Dr Ratna Kumari Svadhyaya Sansthan archived from the original PDF on 20 June 2014 retrieved 19 January 2016 with complete text in Devanagari 110 pages M N Channabasappa 1976 On the square root formula in the Bakhshali manuscript PDF Indian J History Sci 11 2 112 124 David H Bailey Jonathan Borwein 2011 A Quartically Convergent Square Root Algorithm An Exercise in Forensic Paleo Mathematics PDF External links EditThe Bakhshali manuscript MacTutor History of Mathematics archive Ch 6 The Bakhshali manuscript Ian G Pearce Indian Mathematics Redressing the balance Hoernle On the Bakhshali Manuscript 1887 archive org A Big Zero Research uncovers the date of the Bakhshali Manuscript YouTube video University of Oxford Plofker Kim Agathe Keller Takao Hayashi Clemency Montelle and Dominik Wujastyk 2017 The Bakhshali Manuscript A Response to the Bodleian Library s Radiocarbon Dating History of Science in South Asia 5 1 134 50 https doi org 10 18732 H2XT07 Challenges the claims made in the YouTube video A Big Zero Retrieved from https en wikipedia org w index php title Bakhshali manuscript amp oldid 1136094094, wikipedia, wiki, book, books, library,

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