Bakhshali manuscript

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Bakhshali manuscript
Bodleian Library, University of Oxford
Bakhshali numerals 1.png
The numerals used in the Bakhshali manuscript
Type Mathematical text
Date 224–383
Place of origin Bakhshali
Material Birch bark
Format Seventy leaves
Condition Too fragile for examination[1]
Script Śāradā
Discovered 1881

The Bakhshali manuscript is a mathematical text written on birch bark that was found in 1881 in the British-ruled village of Bakhshali (near Mardan in present-day Pakistan). It is notable for being "the oldest extant manuscript in Indian mathematics",[2] with portions dated to AD 224–383. It contains the earliest known Indian use of a zero symbol.[3][4] It is written in Sanskrit with significant influence of local dialects.[2]


The manuscript was unearthed from a field in 1881,[5] by a peasant in the village of Bakhshali, which is near Mardan, now in Khyber Pakhtunkhwa, Pakistan.[2] The first research on the manuscript was done by A. F. R. Hoernlé.[2][6] After his death, it was examined by G. R. Kaye, who edited the work and published it as a book in 1927.[7]

The extant manuscript is incomplete, consisting of seventy leaves of birch bark,[2][5] whose intended order is not known.[2] It is in the Bodleian Library at the University of Oxford[2][5] (MS. Sansk. d. 14), and is said to be too fragile to be examined by scholars.


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.[7][8]


The manuscript is written in an earlier form of Śāradā script, which was mainly in use from the 8th to the 12th century, in the northwestern part of India, such as Kashmir and neighbouring regions.[2] 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.[9] 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.[10] It is possible that the manuscript might be a compilation of fragments from different works composed in a number of language varieties.[9] Hayashi admits that some of the irregularities are due to errors by scribes or may be orthographical.[11]

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.[8] 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.[2]


The manuscript is a compilation of mathematical rules and examples (in verse), and prose commentaries on these verses.[2] 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.[2] 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.[2]

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.[5]

Mathematical context[edit]

Hayasi has compared the text of the manuscript with several Sanskrit texts.[2] 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 Bhasakar[who?]. 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]

The Bakshali manuscript uses numerals with a place-value system, using a dot as a place holder for zero.[12] 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.[13]

Prior to the 2017 carbon dating, a 9th-century inscription of zero on the wall of a temple in Gwalior, Madhya Pradesh, was thought to be the oldest Indian use of a zero symbol.[14]


In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from AD 224–383, 680–779, and 885–993. It is not known how fragments from different centuries came to be packaged together.[15][16][17]

Prior to this finding, most scholars agreed that the physical manuscript was a copy of a more ancient text, whose date had to be estimated based on 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".[7] Channabasappa dated it to AD 200–400, on the grounds that it uses mathematical terminology different from that of Aryabhata.[19] 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".[2]

See also[edit]


  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,[2][7] 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."[18]


  1. ^ All the pages have been photographed which are available in the book by Hayashi
  2. ^ a b c d e f g h i j k l m n o Takao Hayashi (2008), "Bakhshālī Manuscript", in Helaine Selin, Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, 1, Springer, pp. B1–B3, ISBN 9781402045592 
  3. ^ Devlin, Hannah (2017-09-13). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Retrieved 2017-09-14. 
  4. ^ "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 2017-09-14. Retrieved 2017-09-14. 
  5. ^ 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. 
  6. ^ Hoernle 1887.
  7. ^ 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. 
  8. ^ a b Plofker, Kim (2009), Mathematics in India, Princeton University Pres, p. 158, ISBN 978-0-691-12067-6 
  9. ^ a b Hayashi 1995, p. 54.
  10. ^ Section VII 11, corresponding to folio 46v.(Hayashi 1995, p. 54)
  11. ^ Hayashi 1995, p. 26.
  12. ^ Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved 2007-07-24. 
  13. ^ Singh, Maan (1993). Subandhu, New Delhi: Sahitya Akademi, ISBN 81-7201-509-7, pp. 9–11.
  14. ^ "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 2017-09-14. Retrieved 2017-09-14. 
  15. ^ Devlin, Hannah (2017-09-13). "Much ado about nothing: ancient Indian text contains earliest zero symbol". The Guardian. ISSN 0261-3077. Retrieved 2017-09-14. 
  16. ^ Mason, Robyn (2017-09-14). "Oxford Radiocarbon Accelerator Unit dates the world's oldest recorded origin of the zero symbol". School of Archaeology, University of Oxford. Retrieved 2017-09-14. 
  17. ^ "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 2017-09-14. Retrieved 2017-09-14. 
  18. ^ Joseph, G. G. (2000), The Crest of the Peacock, non-European roots of Mathematics, Princeton University Press, pp. 215–216 
  19. ^ E. F. Robinson (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Archived from the original on 9 August 2007. Retrieved 2007-07-24. 


  • 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. 

Further reading[edit]

External links[edit]