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The Bakhshali manuscript, which was unearthed in the 19^{th} century, does not appear to belong to any specific period. Although that said, G Joseph classes it as a work of the early 'classical period', while E Robertson and J O'Connor suggest it may be a work of Jaina mathematics, and while this is chronologically plausible there is no proof it was composed by Jain scholars. L Gurjar discusses its date in detail, and concludes it can be dated no more accurately than 'between 2^{nd} century BC and 2^{nd} century AD'. He offers compelling evidence by way of detailed analysis of the contents of the manuscript (originally carried 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 5^{th} 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).

The controversy and debate surrounding the date of the Bakhshali manuscript was particularly intense when it was first discovered and highlights the resistance of European historians to accept new discoveries and evidence of the origins of various mathematical results. Clearly establishing a date for the composition of the manuscript is extremely important as it has a vast bearing on the significance of its mathematical content. This is in fact the primary requirement for accurate dating of all mathematical works.

The first translation of the manuscript was carried out by G R Kaye, however he was quite unscrupulous in his work and attempted to date the manuscript as 12^{th} Century in order to justify Arabic and Greek influences on the text. Kaye even went as far as to question the Indian origin of the manuscript.

However the vast majority of his translation and suggestions made about the origins of the manuscript have been debunked by less biased (and more accurate) translations. G Joseph further criticises Kaye and comments:

...It is particularly unfortunate that Kaye is still quoted as an authority on Indian mathematics.[GJ, Ps 215-216]

To slightly confuse the issue, it is now considered (almost without doubt) that the manuscript found at Bakhshali is a copy (of the original work) dating from around the 8^{th} century, and certainly no later than 950 AD. The scholar R Hoernle was the first to reach this conclusion following detailed analysis of the manuscript.

I am content to agree with the (prominent) historians who have placed the date at pre 450 AD and identified the 'current' version as a copy. Avoiding further debate, L Gurjar states that the Bakshali manuscript is the:

...Capstone of the advance of mathematics from the Vedic age up to that period.[LG, P49]

Although, as much work was lost between 'periods', we cannot fully gauge continuity of progress and it is possible the composer(s) of the Bakhshali manuscript were not fully aware of earlier works and had to start from 'scratch'. This would make the work an even more remarkable achievement.

The B. Ms. was written on leaves of birch, in Sarada characters and in Gatha dialect, which is a combination of Sanskrit and Prakrit. This may go some way to explaining the number of inaccurate translations. Many of the historians who have been involved in translating ancient Indian works have done so poorly, due to the obscure script, or alternatively because they have not understood the mathematics fully. More worryingly there could be unscrupulous reasons for poor translating in order to play down the importance of ancient Indian works, because they challenge the Eurocentric ideal.

The B. Ms. highlights developments in Arithmetic and Algebra. The arithmetic contained within the work is of such a high quality that it has been suggested:

...In fact [the] Greeks [are] indebted to India for much of the developments in Arithmetic.[LG, P 53]

This quote 'throws open' the traditional Eurosceptic opinion of the history (and origins) of mathematics. Yet even today histories of mathematics rarely acknowledge this contribution of the Indian sub-continent and the B. Ms. is rarely if ever mentioned.

There are eight principal topics 'discussed' in the B. Ms:

Examples of the rule of three (and profit and loss and interest).

Solution of linear equations with as many as five unknowns.

The solution of the quadratic equation (development of remarkable quality).

Arithmetic (and geometric) progressions.

Compound Series (some evidence that work begun by Jainas continued).

Quadratic indeterminate equations (origin of typeax/c=y).

Simultaneous equations.

Fractions and other advances in notation including use of zero and negative sign.

Improved method for calculating square root (and hence approximations for irrational numbers). The improved method (shown below) allowed extremely accurate approximations to be calculated:

A= (a^{2}+r) =a+r/2a- {(r/2a)^{2}/ 2(a+r/2a)}

Example 6.1: Application of square root formula.

Again we can calculate 10, where a = 3 and r = 1.

10 = (32 + 1) = 3 + 1/6 - {(1/36)/2(3 + 1/6)}

= 3 + 1/6 - {(1/36)/(19/3)}

= 3 + 1/6 - 1/228

= 3.16228... in decimal form

10 = 3.16228 when calculated on a calculator and rounded to five decimal places.

Example 6.2: Quadratic equation as found in B. Ms.

If the equation given is

dn^{2}+ (2a-d)n-2s= 0

Then the solution is found using the equation:

n= (- (2a-d) (2a-d)^{2}+8ds))/2d

Which is the quadratic equation witha=d,b= 2a-d, andc= 2s.

Example 6.3: Linear equation with 5 variables.

The following problem is stated in the B. Ms:

"Five merchants together buy a jewel. Its price is equal to half the money possessed by the first together with the money possessed by the others, or one-third the money possessed by the second together with the moneys of the others, or one-fourth the money possessed by the third together with the moneys of the others...etc. Find the price of the jewel and the money possessed by each merchant.Solution.

We have the following systems of equations:

x_{1}/2 +x_{2}+x_{3}+x_{4}+x_{5}=p

x_{1}+x_{2}/3 +x_{3}+x_{4}+x_{5}=p

x_{1}+x_{2}+x_{3}/4 +x_{4}+x_{5}=p

x_{1}+x_{2}+x_{3}+x_{4}/5 +x_{5}=p

x_{1}+x_{2}+x_{3}+x_{4}+x_{5}/6 =p

Then ifx_{1}/2 +x_{2}/3 +x_{3}/4 +x_{4}/5 +x_{5}/6 =qthe equations become (^{377}/_{60})q=p.

A number of possible answers can be obtained. This is the origin of the indeterminate equation of the typeax/c=y, the theory of which was greatly developed, and later perfected by Bhaskara II, four hundred years before it was discovered in Europe.

Ifq= 60 thenp= 377 andx_{1}= 120,x_{2}= 90,x_{3}= 80,x_{4}= 75 andx_{5}= 72

With regard to my central discussion of the neglect of Indian mathematics I feel it is important to highlight the presence of moderately advanced algebra in the B. Ms. Historians of mathematics debate whether true algebra 'began' in Greece or Arabia, and little mention is ever made of Indian algebra. In light of my own research I feel that early Arabic algebra (c. 800 AD) in no way *surpasses* the level of understanding of 6^{th} century Indian scholars.

The Bakhshali manuscript is a unique piece of work and while it not only contains mathematics of a remarkably high standard for the time period, also, in contrast to almost all other Indian works composed before and after, the method of the commentary follows a highly systematic order of:

i. Statement of the rule (

sutra)

ii. Statement of the examples (udaharana)

iii. Demonstration of the operation (karana) of the rule.

The work is considerably less concise than other Indian works which were often (if not always) written in a poetic form comprising of short statements of rules, and rarely included examples. This poetic form was favoured, not only as it gave the authors an opportunity to demonstrate their skills but also because of the limited supplies of writing equipment available.

The reasons for the composition of the B. Ms. are unknown but it seems possible that the motivation was to "bring out" the developments of mathematics during the time period. Thus it seems very likely it was composed for solely academic, intellectual and interest ends, in short, mathematics for mathematics sake.

By the end of the 2^{nd} century AD mathematics in India had attained a considerable stature, and had become divorced from purely practical and religious requirements, (although it is worth noting that over the next 1000 years the majority of mathematical developments occurred within works on astronomy). The topics of algebra, arithmetic and geometry had developed significantly and it is widely thought that the decimal place value system of notation had been (generally) perfected by 200 AD, the consequence of which was far reaching.

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