Bakhshali manuscript and Ramanujam: Mathematical manuscripts from Bodleian Library of Oxford to Cambridge!

Bakhshali manuscript and Ramanujam: Mathematical manuscripts from Bodleian Library of Oxford to Cambridge!

Hard work and esoteric wisdom: Srinivasa Ramanujan claimed that he got such amazing mathematical wisdom from “Namagiri Devi,”[1] just like Kalidasa getting knowledge from Kali. According to Ramanujan, she appeared to him in visions, proposing mathematical formulas and he would have taken down. One such event was described by him as follows:

“While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing“.

 Or it could be taken down in this way also – as he always used to think about such theorems, formulae etc., a sudden spark would have helped him to get the result. Of course, for such intuition, a divine blessing is required. Furthermore, Ramanujan’s mother received permission from Namagiri Thayar for Ramanujan to go to England in a dream. Ramanujam like any other devout Hindu student would have always attributed his knowledge, timely recollection at the time of examination, thereby writing examinations well, getting high marks, etc., only to his favourite God or Goddess. However, his hard work plays a key role always. Thus, definitely, their knowledge was tremendous, stupendous and great only. Such mathematical knowledge was preserved in India historically. Actually, it was not known how many pages the manuscript bound contained, how many recovered, left at the site itself, damaged and cured for research, and kept at the Bodleian library now. Thus, the oldest mathematical manuscript is now known as “Bakhshali manuscript”  contained the Indian mathematical past historically.

Bakhshali manuscript – the notebook/workbook of an Ancient Indian Student: The “Bakhshali manuscript” is nothing but, a notebook or workbook of an ancient Indian student containing many pages, but, only 70 were reportedly recovered in a mutilated condition found between stones, when a peasant living enclosure was dug. In May 1881, near a village called Bakhshali, lying in the Yusufzai district of the Peshawer division, at the extreme Northwestern frontier of India. About its nature also, different versions are available.

The letter dated 5th of July 1881, from the Assistant Commissioner of  Mardan states[2], “According to the finder’s statement the greater part of the manuscript had been destroyed in taking it up from the place where it lay between stones. The remains, when brought to me, were like dry tinder, and there may be about fifty pages left some of which would be certainly legible to anyone who knew the characters. The letters on some of the pages are very clear and look like some kind of Prakrit, but it is most difficult to separate the pages without injuring them. I had intended to forward the manuscript to the Lahore Museum in the hope that it might be sent on thence to some scholar, but I was unable to have a proper tin box made for it before I left Mardan. I will see to this on my return from leave. The papyrus will require very tender manipulation. The result will be interesting, if it enables us to judge the age of the ruins where the manuscript was found.”

Rudolf Hoernle noted, “Unfortunately, probably through the careless handling of the finder, it is now in an excessively mutilated condition, both with regard to the size and the number of the leaves. Their present size, as you observe (see Plate I), is about 6 by 3 ½ inches; their original size, however, must have been about 7 by 8 ¼ inches. This might have been presumed from the well-known fact that the old birch-bark manuscripts were always written on leaves of a squarish size. But I was enabled to determine the point by a curious fact.”

Its size varies from 13×7 to 18×21 cms size. In other words, the sizes of the notebook pages were thus equivalent to A4 to a bigger size. Professor Buhler, who had read of the discovery in the Bombay Gazette communicated the announcement to Professor Weber, who brought it to the notice of the fifth International Congress of Orientalists then assembled in Berlin. In Buhler’s letter to Weber it was stated that the manuscript had been found, “..carefully enclosed in a stone chamber……,” however, the 70 leaves recovered showed that they were damaged severely, while removing. As noted, the “Bakshali manuscript” was nothing but, a student mathematical book cum workbook containing many pages. It has many interesting problems, but, difficult to solve. The manuscript is written in Sharada character of a rather ancient type, in Sanskrit and on the leaves of birch-bark which from age have become dry like tinder and extremely fragile[3].

The tradition of Birch-bark manuscripts of North India spreading from Central Asia to China: The Birch-bark manuscripts are documents written on pieces of the inner layer of birch bark (of tree), which was commonly used for writing in ancient India. The evidence of birch bark for writing goes back many centuries and in various cultures. The oldest dated birch bark manuscripts are numerous Gandhāran Buddhist texts from approximately the 1^st century BCE to CE, which are believed to have created in Gandhara part of Bharat (now Afghanistan), likely by the Dharmaguptaka sect of Buddhists. That the Buddhism was existed till the medieval period above Bharat from Central Asia to China and of course down to South East Asia is well known. In other words, just like the tradition of “Talapatra” of south India, the north Indians used “Bhiojpatra.” Moreover, the writing was there, during the first centuries with an established educational system. As these manuscripts withstood time, there could have been more, but, unless, the Europeans reveal, Indian researchers are not in a position to proceed further. Now coming to numbers and zero, as this Bakshali manuscript contained zeros, it would have been more interesting, had the manuscript was shown to Ramanujam by his British friends of the Cambridge.

The number “0”, decimal numbers etc: According to the westerners / the European experts, the origin of the symbol zero has long been one of the world’s greatest mathematical mysteries. However, The Rigveda Mandala X has the concepts of 0 and 1, giving examples of the concept of “non-existence” and “existence”. Pingala[4] has been credited with the inventor of Binary Mathematics[5]. Here, the Bakshali manuscript contains many zeros mentioned specifically. Though, many consider the symbol noted as dot, big dot etc., it appears as circle only[6]. Therefore, it’s dating from the first centuries was disputed and taken to the 4^th century CE. In 2017, the Bakhshali manuscript, was subjected to radiocarbon dating conducted. Various dates have been obtained ranging from the 3^rd or 4^th century CE to 12^th century. In fact, composed of material from at least three different periods, as per the C-14 dating.

Table[7] Laboratory codes, radiocarbon determinations, stable carbon isotope composition and calibrated ages for the birch bark from the Bakhshali manuscript. Presented uncertainties in the radiocarbon determination are one standard deviation.

Sample	ORAU laboratory code	Radiocarbon determination / BP	δ / ‰	Calibrated age, 95.4% Confidence interval / cal AD
Bakhshali folio 16	OxA-35,405	1751 ± 29	-27.3	224 – 383
Bakhshali folio 17	OxA-35,406	1247 ± 27	-27.0	680 – 868
Bakhshali folio 33	OxA-35,407	1108 ± 26	-24.6	885 – 993

If that is the case, no student or scholar would keep his book, notebook or textbook with such mixed manuscripts. Such mix-up must have done by the discoverer of or the person recovered the manuscripts. Probably, they could have recovered three such manuscripts at different levels. As Dakshasila University was nearby and it was destroyed by the Mohammedans, all these books could have been found scattered. Or the “enclosed” place, where, the manuscript was found, could have been a school that was destroyed. The most elaborate and authoritative academic study on the manuscript, conducted by Japanese scholar Dr Hayashi Takao[8], asserted that it probably dated from between the 8^th and the 12^th century, based on factors such as the style of writing and the literary and mathematical content. Another researcher has noted some astronomical details also[9]. Thus, the earlier dating of the manuscript goes before the 9th-century inscription of zero found on the wall of a temple in Gwalior, Madhya Pradesh, which was previously considered to be the oldest recorded example of a zero used as a placeholder in India.

The zero symbol could have been evolved from a dot that was used in ancient India and can be seen throughout the Bakhshali manuscript. The dot was originally used as a ‘placeholder’, meaning it was used to indicate orders of magnitude in a number system – for example, denoting 10s, 100s and 1000s. In fact, in the Bakhshali Mss, it is clearly written as 0 instead of a big dot.  The “Number system” that was reportedly used by the Egyptian, Babylonian, Mayan and Roman are compared with that of the Indian for comparison. While the use of zero as a placeholder was seen in several different ancient cultures, such as among the ancient Mayans and Babylonians, the symbol in the Bakhshali manuscript is particularly significant for two reasons. Firstly, it is this dot that evolved to have a hollow centre and became the symbol that we use as zero today. Secondly, it was only in India that this zero developed into a number in its own right, hence creating the concept and the number zero that we understand today – this happened in 628 CE, just a few centuries after the Bakhshali manuscript was produced, when the Indian astronomer and mathematician Brahmagupta wrote a text called Brahmasphutasiddhanta, which is the first document to discuss zero and its operations.

Ramanujam and zero: The concept and evolution of 0 and ∞ haven challenging for all civilizations, but, the ancient Indians were dealing with them, philosophically and mathematically with clear-cut logic. Of course, there was no difference between philosophy and mathematics till medieval periods. Ramanujam asked his teacher what would happen if zero was divided by zero. They did not use 0 till 12^th century and ∞ 17^th century[10]. The westerners did not recognize them till the medieval / modern period. If any number is divided by itself, 1 is obtained, then, zero divided by zero would give 1 as result. It is not known as to Ramanujan would have read “Bija ganita” of Bhaskaracharya or not. However, it is evident that he had taken much interest in zero and other numbers. Thus, had Ramanujan seen the Bakashali manuscript, many riddles of the manuscript would have been solved. After all, the distance between Cambridge and Oxford is about 100 kms and it would take two hours to reach. It is not known who prevented him not to see the Bakhshali manuscript.

© K. V. Ramakrishna Rao

27-04-2020

[1] Ramanujan prayed to the goddess Namagiri by sitting in the center of a four pillared mandapam facing the goddess, in the Narasimha swamy Temple. It is said that they stayed in the precincts of the temple for three days, and Ramanujan got the permission of the goddess to go to England, in a dream when he was asleep. He woke Narayana Iyer and told him that his mission of getting the permission of the goddess to go to England was accomplished.

[2] G. R. Kaye, The Bakhshali Manuscript – A Study in Medieal Mathematics, Vol.I, in Indian Historical researches, Royal Asiatic Society, Calcutta, 1933,  p.1

[3] Birch-bark is an outer bark of the Silver Birch (Betula utilis, Betula bhojpaltra, or the Bhurja tree, as it is vai’iously called) which flourishes in the Himalayas from Kashmir to Sikkim. It grows on all the higher ranges of the Kashmiri- hills from a height of about 6,000 feet to 12,000 foot. The forests in the (Gurais district supply most of the bhojpatra that is sold in Srinagar. The bark is used chiefly for the roofing of houses, for wrapping up things, for lining baskets, etc., and the villagers still use it as a writing material.

1. R. Kaye, Vol.I, p.4

[4] Pingala (c.3rd / 2nd century BCE) was an ancient mathematician, famous for his work, the Chandas shastra (chandas-śāstra, also Chandas sutra chandas-sūtra), a Sanskrit treatise on prosody considered one of the Vedanga. Pingala is identified as the younger brother of Panini, the great grammarian of c. 5^th century BCE. Other traditions identify him with Patanjali, the author of the Mahabhashya.

[5] K. V. Ramakrishna Rao, Date of Pingala – The Origin of Binary Computation in India, VYOMA, Bangalore, 2004, pp.356-362.

[6] R.N. Mukherjee; Zero, Encyclopaedia of Classical Indian Sciences, H. Selin, R. Narasimha (edt.), Universities Press, (2007) pp. 441-444.

[7] David Howell, Carbon dating reveals Bakhshali manuscript is centuries older than scholars believed and is formed of multiple leaves nearly 500 years different in age, By David Howell, Head of Heritage Science at the Bodleian Libraries, 3 July 2017

[8] Hayashi Takao, The Bakhshali Manuscript – an ancient mathematical tratise, Egbert Forsten, Groningen, Netherlands,1995.

[9] Syshma Zadoo, Critical study of the Bakhshali Manuscript, PhD thesis, University of Kashmir, 1992, pp.141-142.

[10] K. V. Ramakrishna Rao, From Zero to Infinity, in “Vigyan Bharati Pradeepika”, Bharatiya Vigyan Sammelan, – Proceedings, Vol.8, No.1, April 13, 2002,  pp. 45-54.