Bakhshali manuscript

Spelling of the numbers in the Bakhshali manuscript

The Bakhshali Manuscript is a manuscript containing a collection of mathematical scriptures written on birch bark . It was found in 1881 near the eponymous village of Bakhshali (today in Pakistan , 80 km northeast of Peshawar ). Today it is in the Bodleian Library in Oxford. The dating of the manuscript is controversial. What is certain is that it is the oldest surviving mathematical manuscript in India. It is possible that the Bakhshali manuscript also contains the oldest material evidence of the use of the zero (here represented by a point) in the Indian region.

There is little evidence of the author's background, but it suggests a Brahmanic origin.

Description and dating

The manuscript is incomplete, consists of 70 fragmentary leaves of birch bark and is written in the Sharada script , which was used in Kashmir until the 12th century . The language is Sanskrit (influenced by the regional dialects). The original order of the leaves is uncertain. Editions appeared in 1887 by Rudolf Hoernlé and later by GR Kaye.

The Bakhshali manuscript is considered to be the oldest surviving mathematical manuscript in India. The dating is controversial. According to Hoernlé it dates from the 3rd to 4th centuries AD (and also according to Bibhutibhushan Datta from the first centuries AD), GR Kaye places it in the 12th century, David Pingree and Ayyangar in the 8th to 9th centuries. Since it has similarities in the form of the problems and in the terms used with the commentary by Bhaskara I on Aryabhata , Takao Hayashi classified it in the same period (7th century); he placed the manuscript itself in the period from the 8th to 12th century.

In 2017, a radiocarbon study of three sheets of the Bakhshali manuscript was carried out. According to the results of the investigation, all three leaves come from different centuries. The oldest was dated to AD 224–383, one to AD 680–779 and one to AD 885–993. Accordingly, the oldest parts of the Bakhshali manuscript would be significantly earlier than previously assumed. In a reply, Kim Plofker , Takao Hayashi and colleagues consider it likely that even the oldest parts were only described at the time of the later dating. Because of the uniform font and the unity of the text, it is highly unlikely that the production of the individual parts of the manuscript would be so far apart. If the results of the radiocarbon dating are correct, it must be assumed that the date of the most recent sheet determines the time of writing. They also pointed out that there had already been an example of radiocarbon dating of a Tibetan manuscript that let the parts fall apart 600 years, while the philological, historical and art-historical analysis spoke for a unified origin, and criticize the high publicity of the radiocarbon dating the Bodleian Library without first attempting a scientific review process.

content

The work mainly deals with arithmetic and algebra . It probably has different authors, one is called the son of Chajaka, who in turn is said to have been a well-known mathematician and Brahmin. It may have been written in the country of Martikavata in northwest India (after a passage in the text that has only been partially preserved).

The manuscript contains a collection of mathematical rules and problems from various sources. The form of the presentation is such that first the rule (sutra) is given, then an example (udaharana) in the form of: problem definition, calculation, verifications. The rules are in verse form with explanations (not in verse form) using examples. In addition to problems that lead to linear equations (including the problem of 100 birds , one should buy 100 birds with 100 coins of different value), it contains a few geometric problems, problems related to travel, and a method for the approximate calculation of the square root:

{\ displaystyle {\ sqrt {(a ^ {2} + r)}} \ approx a + {\ frac {r} {2a}} - {\ frac {{({\ frac {r} {2a}})} ^ {2}} {2 (a + {\ frac {r} {2a}})}}}

notation

A period is used for the zero, with the zero serving as a space character in the place value system. If the results of the radiocarbon dates published in 2017 are correct, the Bakshali manuscript is the oldest evidence of zero in India. Assuming a uniform origin of the manuscript, there are, according to Plofker and colleagues, indications of the independent use of zero as a number, since multiplications of long numbers occur in decimal notation, which suggest an algorithm that precedes digit by digit and thus also arithmetic operations for the zero provided. At another point a one was explicitly added to zero. Negative numbers (subtraction) are indicated by a trailing “plus” (in later Indian mathematical works a point is often placed above the number for this).

literature

Rudolf Hoernlé: On the Bakshali manuscript, Vienna: A. Hölder 1887 (negotiations of the VII International Congress of Orientalists, Vienna 1886), archive
Rudolf Hoernlé: The Bakshali Manuscript, The Indian Antiquary, Volume 17, 1888, pp. 33-48, 275-279
Takao Hayashi: "Bakhshālī Manuscript", in: Helaine Selin, Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer 2008
Takao Hayashi: The Bakshali Manuscript, an ancient Indian mathematical treatise, Groningen: E. Forsten 1995
Takao Hayashi: Indian Mathematics, in: Gavin Flood, The Blackwell Companion to Hinduism, Blackwell 2003, pp. 360-375
GR Kaye: The Bakshali Manuscript: a study in medieval mathematics, Archaeological Survey of India, Part 1,2, Calcutta 1927, Part 3 Delhi 1933, Reprint New Delhi 1981, 1987
GR Kaye: The Bakshali Manuscript, Journal of the Asiatic Society of Bengal, Volume 8, 1912, pp. 349-361
AAK Ayyangar: The Bakshali Manuscript, Mathematics Student, Volume 7, 1939, pp. 1-16
B. Datta : The Bakshali Mathematics, Bulletin of the Calcutta Mathematical Society, Volume 21, 1929, pp. 1-60
David Pingree : Census of Exact Sciences in Sanskrit, 5 volumes, Philadelphia 1970 to 1994
Svami Satya Prakash Sarasvati, Usha Jyotishmati: The Bakshali Manuscript. An Ancient Treatise on Indian Arithmetics, Allahabad 1979, pdf
MN Channabasappa: On the square root formula in the Bakhshali manuscript, Indian J. History Sci., Vol. 2, 1976, pp. 112-124
Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle, Dominik Wujastyk: The Bakshali Manuscript: A response to the Bodleian Library's Carbon Dating , History of Science in South Asia, 5.1, 2017, pp. 134–150

Web links

McTutor on the Bakshali manuscript

Individual evidence

↑ Plofker, Keller Hayashi et al. a., The Bakshali Manuscript. A response to the Bodleian Library`s Radiocarbon Dating, History of Science in South Asia 2017, p. 144
↑ Takao Hayashi, article Bakshali Manuscript , In: Helaine Selin, Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer 2008
↑ For example, ya (abbreviation of yavattavat for as much as ) is not used systematically as in Bhaskara, but only occasionally for unknowns in algebraic equations
↑ Plofker, Keller Hayashi et al. a., The Bakshali Manuscript. A response to the Bodleian Library`s Radiocarbon Dating, History of Science in South Asia 2017, p. 135
↑ Hannah Devlin: Much ado about nothing: ancient Indian text contains earliest zero symbol. The Guardian, September 14, 2017, accessed September 14, 2017 . Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'. Bodleian Library, September 14, 2017, accessed September 14, 2017 .
↑ Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle, Dominik Wujastyk: “The Bakhshālī Manuscript: A Response to the Bodleian Library's Radiocarbon Dating”, in: History of Science in South Asia 5.1 (2017), pp. 134–150. doi: 10.18732 / H2XT07 .
↑ Plofker, Keller, Hayashi et al. a., 2017, p. 139
↑ Plofker, Hayashi et al. a., 2017, p. 140. Marcus du Sautoy had previously claimed in the context of the announcement of the radiocarbon dating of the Bodleian Library that there was no reference to the use of the zero as an independent number, but only in a placeholder function (such a use as a placeholder it was already used by the Babylonians and the knowledge of a similar use in India in the first centuries after Christ would not be a surprise, according to Plofker and colleagues).

[1] Plofker, Keller Hayashi et al. a., The Bakshali Manuscript. A response to the Bodleian Library`s Radiocarbon Dating, History of Science in South Asia 2017, p. 144

[2] Takao Hayashi, article Bakshali Manuscript , In: Helaine Selin, Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer 2008

[3] For example, ya (abbreviation of yavattavat for as much as ) is not used systematically as in Bhaskara, but only occasionally for unknowns in algebraic equations

[4] Plofker, Keller Hayashi et al. a., The Bakshali Manuscript. A response to the Bodleian Library`s Radiocarbon Dating, History of Science in South Asia 2017, p. 135

[5] Hannah Devlin: Much ado about nothing: ancient Indian text contains earliest zero symbol. The Guardian, September 14, 2017, accessed September 14, 2017 . Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'. Bodleian Library, September 14, 2017, accessed September 14, 2017 .

[6] Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle, Dominik Wujastyk: “The Bakhshālī Manuscript: A Response to the Bodleian Library's Radiocarbon Dating”, in: History of Science in South Asia 5.1 (2017), pp. 134–150. doi: 10.18732 / H2XT07 .

[7] Plofker, Keller, Hayashi et al. a., 2017, p. 139

[8] Plofker, Hayashi et al. a., 2017, p. 140. Marcus du Sautoy had previously claimed in the context of the announcement of the radiocarbon dating of the Bodleian Library that there was no reference to the use of the zero as an independent number, but only in a placeholder function (such a use as a placeholder it was already used by the Babylonians and the knowledge of a similar use in India in the first centuries after Christ would not be a surprise, according to Plofker and colleagues).