Posted on: 29 January 2015

Digital Rare Book:
On the Bakhshali Manuscript
By Dr. Rudolf Hoernle
Published by Alfred Holder, Vienna - 1887

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The Bakhshali manuscript is an early mathematical manuscript which was discovered over 100 years ago. The Bakhshali Manuscript is the name given to the mathematical work written on birch bark and found in the summer of 1881 near the village Bakhshali (or Bakhshalai) of the Yusufzai subdivision of the Peshawar district (now in Pakistan). The village is in Mardan tahsil and is situated 50 miles from the city of Peshawar.

An Inspector of Police named Mian An-Wan-Udin (whose tenant actually discovered the manuscript while digging a stone enclosure in a ruined place) took the work to the Assistant Commissioner at Mardan who intended to forward the manuscript to Lahore Museum. However, it was subsequently sent to the Lieutenant Governor of Punjab who, on the advice of General A Cunningham, directed it to be passed on to Dr Rudolf Hoernle of the Calcutta Madrasa for study and publication. Dr Hoernle presented a description of the BM before the Asiatic Society of Bengal in 1882, and this was published in the Indian Antiquary in 1883. He gave a fuller account at the Seventh Oriental Conference held at Vienna in 1886 and this was published in its Proceedings. A revised version of this paper appeared in the Indian Antiquary of 1888. In 1902, he presented the Bakhshali Manuscript to the Bodleian Library, Oxford, where it is still (Shelf mark: MS. Sansk. d. 14).

A large part of the manuscript had been destroyed and only about 70 leaves of birch-bark, of which a few were only scraps, survived to the time of its discovery.

To show the arguments regarding its age we note that F R Hoernle, referred to in the quotation above, placed the Bakhshali manuscript between the third and fourth centuries AD. Many other historians of mathematics such as Moritz Cantor, F Cajori, B Datta, S N Sen, A K Bag, and R C Gupta agreed with this dating. In 1927-1933 the Bakhshali manuscript was edited by G R Kaye and published with a comprehensive introduction, an English translation, and a transliteration together with facsimiles of the text. Kaye claimed that the manuscript dated from the twelfth century AD and he even doubted that it was of Indian origin.

Channabasappa gives the range 200 - 400 AD as the most probable date. In the same author identifies five specific mathematical terms which do not occur in the works of Aryabhata and he argues that this strongly supports a date for the Bakhshali manuscript earlier than the 5th century. Joseph in suggests that the evidence all points to the:-

... manuscript [being] probably a later copy of a document composed at some time in the first few centuries of the Christian era.

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A leaf from a birch bark manuscript that was discovered in 1881 by a farmer while digging a field about fifty miles from modern Peshawar in Pakistan. The text is a collection of algorithms and sample problems in verse, with a commentary explaining them in a combination of prose and numerical notation. It provides unique evidence for how ancient Indian mathematics was written in manuscript. Numerals are expressed in decimal place form and the zero is represented by a round dot. Quantities are expressed in numerals set off from the text by horizontal and vertical boxes. Fractions are written in the familiar way, but with no line dividing numerator and denominator, and negative values are shown by a small cross after the number, similar to the modern ‘+’ sign.

© 2011 Bodleian Libraries

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Another interesting piece of mathematics in the manuscript concerns calculating square roots. The following formula is used √Q = √(A2 + b) = A + b/2A - (b/2A)2/(2(A + b/2A)) This is stated in the manuscript as follows:- In the case of a non-square number, subtract the nearest square number, divide the remainder by twice this nearest square; half the square of this is divided by the sum of the approximate root and the fraction. this is subtracted and will give the corrected root. Taking Q = 41, then A = 6, b = 5 and we obtain 6.403138528 as the approximation to √41 = 6.403124237. Hence we see that the Bakhshali formula gives the result correct to four decimal places. The Bakhshali manuscript also uses the formula to compute √105 giving 10.24695122 as the approximation to √105 = 10.24695077. This time the Bakhshali formula gives the result correct to five decimal places. The following examples also occur in the Bakhshali manuscript where the author applies the formula to obtain approximate square roots: √487 Bakhshali formula gives 22.068076490965 Correct answer is 22.068076490713 Here 9 decimal places are correct √889 Bakhshali formula gives 29.816105242176 Correct answer is 29.8161030317511 Here 5 decimal places are correct [Note. If we took 889 = 302 - 11 instead of 292 + 48 we would get Bakhshali formula gives 29.816103037078 Correct answer is 29.8161030317511 Here 8 decimal places are correct] √339009 Bakhshali formula gives 582.2447938796899 Correct answer is 582.2447938796876 Here 11 decimal places are correct Source: