The Bakhshali manuscript is an ancient Indian mathematical text written on birch bark that was found in 1881 in the village of Bakhshali, Mardan (near Peshawar in present-day Pakistan). 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 Sanskrit with significant influence of local dialects.[4]


The manuscript was unearthed from a field in 1881,[7] by a peasant in the village of Bakhshali, which is near Mardan, now 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 in the Bodleian Library at the University of Oxford[4][7] (MS. Sansk. d. 14), and is said to be too fragile to be examined by scholars.
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][10]


The manuscript is written in an earlier form of Śāradā script, a script which is known for having been in use mainly from the 8th to the 12th century in the northwestern part of India, 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.[11] 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.[12] It is possible that the manuscript might be a compilation of fragments from different works composed in a number of language varieties.[11] Hayashi admits that some of the irregularities are due to errors by scribes or may be orthographical.[13]

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.[10] 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]

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.

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.[14] 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.[15]

Prior to the 2017 carbon dating – which, however, has in the meantime been discounted, see below under Date – 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.[6]

In 2017, three samples from the manuscript were thought to come from three different centuries, from AD 224–383, 680–779, and 885–993, on the basis of a study involving radiocarbon dating. If the dates were accepted, it is not known how fragments from different centuries came to be packaged together.[5][16][6]

A detailed reconsideration of all relevant evidence regarding the date of the Bakhshali manuscript, led Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk to conclude the following: "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. ... we urge the investigators to consider the importance of reconciling their findings with historical knowledge and inferences obtained by other means. It should not be hastily assumed that the apparent implications of results from physical tests must be valid even if the conclusions they suggest appear historically absurd."[1]

Referring to the detailed reconsideration of the evidence by Kim Plofker et al., Jan Houben remarked: "If the finding that samples of the same manuscript would be centuries apart is not based on mistakes in the procedure of sampling etc., or if the manuscript was at the moment it was written upon not partly consisting of older, recycled pages, 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. Among the variables of carbon dates, variation in script and linguistic variation, the first is the most objective but still much in need of calibration for relatively recent, historical dates."[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.[18] 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 Jan Houben observed: "In view of the strong normativity of linguistic usage within the dimension “sanskrit - approximative sanskrit” it is difficult to derive a linear chronological difference from the observed linguistic variation. Also writing is a normative activity and moreover dependent on some amount of individual variation from scribe to scribe. However, writing has been much less subject either to the intensive study of early scripts by later generation scribes or to the conscious reintroduction of archaisms in later forms of writing (something we see in language, most famously the studied archaizing “Vedic” language use in parts of the Mahābhārata and in the Bhāgavatapurāṇa). We therefore have 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. The “hardest” evidence to judge the date of a manuscript such as the Bakhshali and its sections would therefore be the palaeographic evidence. Other evidence, including the laboratory results of radiocarbon dating, is to be interpreted in the light of the results reached by careful palaeographic study."[2]
See also

Birch bark manuscript
Bakhshali approximation
Indian mathematics
Zero (number)


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)

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."[17]


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.
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:
All the pages have been photographed which are available in the book by Hayashi
Takao Hayashi (2008), "Bakhshālī Manuscript", in Helaine Selin (ed.), Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, 1, Springer, pp. B1–B3, ISBN 9781402045592
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.
"Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 2017-09-14. Retrieved 2017-09-14.
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.
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.
Plofker, Kim (2009), Mathematics in India, Princeton University Pres, p. 158, ISBN 978-0-691-12067-6
Hayashi 1995, p. 54.
Section VII 11, corresponding to folio 46v.(Hayashi 1995, p. 54)
Hayashi 1995, p. 26.
Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved 2007-07-24.
Singh, Maan (1993). Subandhu, New Delhi: Sahitya Akademi, ISBN 81-7201-509-7, pp. 9–11.
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. Archived from the original on 2017-09-14. Retrieved 2017-09-14.
Joseph, G. G. (2000), The Crest of the Peacock, non-European roots of Mathematics, Princeton University Press, pp. 215–216

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.
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), The Bakhshali Manuscript: An Ancient Treatise of Indian Arithmetic (PDF), Allahabad: Dr. Ratna Kumari Svadhyaya Sansthan, archived from the original (PDF) on 2014-06-20, retrieved 2016-01-19 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
6 – The Bakhshali manuscript
Hoernle: On the Bakhshali Manuscript, 1887,
"A Big Zero: Research uncovers the date of the Bakhshali Manuscript", YouTube video, University of Oxford

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