The 70 Birch-Bark Leaves of the Bakhshali Manuscript Hold Hundreds of Dot Zeros, and Oxford's Latest Dating Puts the Bark of Five Leaves Between 773 and 1032 CE

The Bakhshali manuscript is 70 leaves of birch bark covered in Sanskrit rules and worked arithmetic problems, with a dot for zero written hundreds of times across them. The Oxford radiocarbon laboratory and the Bodleian Libraries, which hold it, call it “almost certainly the oldest extant witness of a South Asian mathematical work” in their 2024 report. That report dates the bark of five of its leaves to between 773 and 1032 CE.

The date changed once already. In September 2017 the same institutions announced that one leaf was as old as the 3rd or 4th century, and The Guardian and other outlets reported it. The 2024 report explains why that figure no longer stands. The text and its zeros are unchanged; only the measured age of the bark moved.

A labourer, a ruin and a village near Peshawar

The leaves were found in May 1881 at Bakhshali, a village about 80 kilometres north-east of Peshawar in what is now Pakistan. The Oxford report says an agricultural labourer recovered them from either a field or a ruin, and that the accounts of the find are not precise.

A report by the German scholar A. F. R. Hoernle, then Government Palaeographer of India, from 1882 says the manuscript was found in a ruined enclosure by a man digging for stones. Hoernle adds that most of it was destroyed when the finder took it up. He studied the surviving pieces and in 1902 presented them to the Bodleian Library in Oxford, where they are kept under the shelfmark MS. Sansk. d. 14.

Most of what scholars know about the text still comes from photographs. G. R. Kaye published them with his edition in 1927. Afterwards, according to the mathematician Bill Casselman of the University of British Columbia, the fragments were encased in mica sheets. The bark stuck to the mica and broke into hundreds of tiny pieces, and it darkened. Casselman notes that it is apparently still possible to photograph the pages well through the mica.

What the leaves teach

The text is a compendium of rules and sample problems, written in verse with a prose commentary. The Oxford report describes it as “algorithms and sample problems in verse” in a Sanskrit influenced by regional languages of the north-west. The script is a Sharada variant, and the Würzburg Indoskript database labels the folio it lists as Late Proto-Sharada. In the pattern Casselman describes, each rule is followed by examples and then a check of the answer.

The problems are everyday ones. A mixture problem on folios 16 and 17 deals with gold alloys of different purity. Others, MacTutor reports, cover the rule of three (its example works out 50 times 12 divided by 8 as 75 dinaras), two boys whose daily wages and debts leave them level after 30 days, and an indeterminate problem about the price of horses. Marcus du Sautoy, the Oxford mathematician, summarised the flavour in 2017 as questions of the kind “If someone buys this and sells this how much have they got left?”

The square roots are the showpiece. MacTutor sets out the manuscript’s method for numbers that are not perfect squares and gives its value for the square root of 41 as 6.403138528, against a true value of 6.403124237, correct to four decimal places. For 339,009 it gets eleven decimal places right. Casselman says he knows of nothing like its very large numbers in other cultures until much later, and notes that decimal fractions were invented by Arab mathematicians only after this kind of work.

The colophon, the scribe’s note at the end, names a Brahmin described as the son of Chajaka, who wrote the work for one Hasika, son of Vasiha, and his descendants, in a place in the Gandhara region of north-west South Asia. Takao Hayashi edited and translated the text in 1995, and the 2017 reply discussed below, of which he is a co-author, calls that edition the definitive study. He tentatively dated the commentary to the 7th century, and the surviving manuscript to some time between the 8th and 12th centuries.

The dot that does two jobs

The zero in the manuscript is a dot. It marks an empty place in a number, as the 0 does in 101, and the 2017 Oxford announcement said it later grew a hollow centre and became the modern symbol. The authors of the 2017 reply add that the manuscript also uses the dot for an unknown quantity, in a way that resembles a variable in algebra.

The rules also show zero being computed with. In a problem on folio 22 the dot standing for an unknown is replaced by 1, and the text says the step is done “having added unity to zero”. On folio 47 the sample problem multiplies the fractions 108625/65600 and 59425/49200 and gives 6455040625/3227520000, a calculation that the five authors say is hard to picture without arithmetic on the zeros inside the numbers. Oxford’s 2017 release described the Bakhshali dot as a placeholder and named Brahmagupta’s Brahmasphutasiddhanta of 628 CE as the first document to discuss zero as a number. The rules Brahmagupta laid down are covered in their own article.

The wider history of the symbol is in India gave the world zero. The Bakhshali leaves are one piece of evidence in it, and the 2017 reply lists others: the Sanskrit word shunya (empty) served some notational purpose in calculations on verse metres, a text from around 550 CE defines a constant as “sixty minus zero”, and several early texts describe place-value counting on rods.

September 2017: three leaves, three centuries

The Bodleian had three small samples of bark taken, away from the ink, from folios 16, 17 and 33. The Oxford Radiocarbon Accelerator Unit, working with Dr David Chivall, measured them. Camillo Formigatti, the Bodleian’s John Clay Sanskrit Librarian, posted the ranges to a scholars’ mailing list on 14 September 2017, each at 95.4 per cent confidence: 224 to 383 CE for folio 16, 680 to 779 CE for folio 17 (with a smaller second interval, 790 to 868 CE), and 885 to 993 CE for folio 33. He wrote that the team had not expected such a big difference between the three.

The University of Oxford released the news that day and The Guardian ran it as an exclusive. The release said the manuscript was as old as the 3rd or 4th century, about five centuries earlier than scholars believed, and older than the 9th-century zero on a temple wall in Gwalior, Madhya Pradesh, until then the oldest recorded Indian placeholder zero. It said the leaves came from at least three different periods. The release named Hayashi’s 8th to 12th century range as the leading earlier academic estimate.

The scholars’ reply

In 2017 five historians of Indian mathematics (Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk) published a response in the peer-reviewed journal History of Science in South Asia. They said they could not challenge the radiocarbon figures on technical grounds, and argued from the manuscript itself. The handwriting and layout look uniform across the leaves, which in their view does not fit pages written centuries apart. The back of folio 16 and the front of folio 17 carry one continuous problem on mixing gold alloys, and Hayashi saw no sign of a different scribe. If the work was written out as a single piece, they wrote, the date of the zeros is the date of the youngest leaf.

Their conclusions also credit the testing. They called radiocarbon analysis potentially of great value and asked for more than one sample per leaf and, if possible, tests on the inked letters themselves. They regretted that the results had gone to the press and a video before peer review.

On the mailing list the next day, Walter Slaje wrote that the early Indian use of zero was long-established knowledge among Indologists and asked why none had been consulted. Andrea Acri noted that if the manuscript were a single text its copying date would fall in the 9th or 10th century, and said he was working from the press report without access to a library.

Oxford’s 2024 report

On 14 October 2024 the Oxford Radiocarbon Accelerator Unit and the Bodleian deposited a fuller report by David Chivall, Virginia Lladó-Buisán, David Howell and Gillian Evison. It says the 2017 measurements on folios 16 and 17 produced unexpected ages, so the team ran new tests in 2018 on folios 23 and 55, on folio 15, and on a second sample from folio 16. The second folio 16 result, the report states, showed the first had been inaccurate. Folio 17’s measurement was an outlier compared with the rest and was left out. Folio 33’s result was consistent with the new ones and stayed in.

The five calibrated ranges, at 95.4 per cent, are 773 to 986 CE for folio 15, 901 to 1032 for folio 16, 890 to 1014 for folio 23, 885 to 995 for folio 33 and 774 to 991 for folio 55. The Würzburg Indoskript database lists folio 16 at 901 to 1032 CE on the basis of radiocarbon dating, in line with the new result. The report’s statistical tests found that the five leaves cannot all be assumed to have the same age, though the four other than folio 16 can, and these give 889 to 979 CE. A model that makes no assumptions about how the leaves relate puts the start of the manuscript’s production in 799 to 892 CE and the end in 900 to 1102 CE.

The report is careful about what these numbers date. Radiocarbon measures when the bark was formed, and the manuscript cannot have been written before that, so the dates set the earliest possible time for its making.

What stands

The same report says proposed dates for the manuscript have ranged from about 200 CE to 1100 CE, and that it is almost certainly the oldest surviving witness of a South Asian mathematical work. Casselman calls it by far the oldest surviving mathematics manuscript and says birch bark does not last long in a damp climate.

The Gwalior inscription that Oxford’s 2017 release called the oldest Indian placeholder zero is dated to the 9th century, and the new bark ranges overlap that century. Hayashi’s 7th-century date for the commentary comes from handwriting, language and the history of Sanskrit arithmetic texts, and the 2017 reply’s reading of the zeros as numbers rests on the text’s own procedures. Neither is a measurement of the bark.

Six of the 70 leaves have been sampled (folios 15, 16, 17, 23, 33 and 55). The Oxford report does not expect further dating, because each test uses up a small sample of the bark.