The Bakhshali Manuscript is a mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881. It is notable for being "the oldest extant manuscript in Indian mathematics."
The Bakhshali manuscript is incomplete, with only seventy leaves of birch bark, many of which are mere scraps. Even the intended order of the 70 leaves is indeterminate. It is currently housed in the Bodleian Library at the University of Oxford (MS. Sansk. d. 14) and is currently too fragile to be examined by scholars.
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. The language is the Gatha dialect (which is a combination of the ancient Indian languages of Sanskrit and Prakrit).
A colophon to one of the sections, that says:
This has been written by the son of Chajaka, a brāhmaṇa and king of mathematicians, for the sake of Hasika, son of Vasiṣṭha, in order that it may be used (also) by his descendants.
is preceded by a broken word rtikāvati, which is believed to be the same as the place Mārtikāvata that is mentioned by Varāhamihira. He mentions this place in his Bṛhatsaṃhitā (16.25) among other locations in northwestern India, such as Takṣaśilā, Gandhāra, etc. Based on this, it is believed that the work of the Bakhshālī manuscript may have been composed in that region.
The manuscript is a compilation of mathematical rules and examples (in verse), and prose commentaries on these verses. Typically, a rule is given, with an example or 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. 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.
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.
Its date is uncertain, and has generated considerable debate. Most scholars agree that the physical manuscript is a copy of a more ancient text, so that the dating of that ancient text is possible only based on the content. Recent scholarship dates it between the 2nd century BC and the 3rd century AD; Ian Pearce summarizes the positions:
Gurjar discusses its date in detail, and concludes it can be dated no more accurately than 'between 2nd century BC and 2nd century AD'. He offers compelling evidence by way of detailed analysis of the contents of the manuscript (originally carried out by R Hoernle). His evidence includes the language in which it was written ('died out' around 300 AD), discussion of currency found in several problems, and the absence of techniques known to have been developed by the 5th century. Further support of these dates is provided by several occurrences of terminology found only in the manuscript, (which form the basis of a paper by M Channabasappa).
However, earlier scholars have tended to date it around 400 AD (Hoernle, Datta/Singh, Bag, Gupta). Hayashi had suggested a possible 7th-century date, while in an early colonial estimate, G.R. Kaye had assessed it to be as late as the 12th century AD. Such late dates are quite unlikely because the language used was already dying by the 4th century; also the work does not mention integer equations and other topics which were of widespread interest after Aryabhata (5th century AD). Today, Kaye's assessment is widely discredited.
The reason why the date of the manuscript is important, is that if the work indeed dates from the 3rd century or earlier, it would imply that the concept of the mathematical zero was known several centuries earlier than the work of Brahmagupta in the 7th century.
- Takao Hayashi (2008), "Bakhshālī Manuscript", in Helaine Selin, Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (Springer) 1: B1, ISBN 9781402045592
- 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.
- Ian Pearce (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Archived from the original on 9 August 2007. Retrieved 2007-07-24.
- T Hayashi, The Bakhshali manuscript: An ancient Indian mathematical treatise (Groningen, 1995).
- Joseph, G. G. (2000). The Crest of the Peacock, non-European roots of Mathematics. Princeton and Oxford: Princeton University Press. Quote: "...It is particularly unfortunate that Kaye is still quoted as an authority on Indian mathematics." [p.215–216]
- Bibhutibhusan Datta (Volume 35, Number 4 (1929), 579–580.). Review: G. R. Kaye, The Bakhshâlî Manuscript—A Study in Mediaeval Mathematics, 1927. Bull. Amer. Math. Soc. Retrieved 2007-07-24.
- M N Channabasappa (1976). "On the square root formula in the Bakhshali manuscript". Indian J. History Sci 11 (2): 112–124
- Augustus Hoernle (1887). "On the Bakshali manuscript"
- David H. Bailey, Jonathan Borwein (2011). "A Quartically Convergent Square Root Algorithm: An Exercise in Forensic Paleo-Mathematics"