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(4 II. Sulba Sutras)
(6: The Bakhshali manuscript)
Following the decline of the Vedic religion around 400BC, the Jaina religion (and Buddhism) became the prominent religion(s) on the Indian subcontinent and gave rise to Jaina mathematics. N Dwary (and others) contend:
...According to the religious literature of the Janias, the knowledge of "Sankhyana" (i.e., the science of numbers, which included arithmetic and astronomy) was considered to be one of the principal accomplishments of Jain priests. [ND1, P 39]
The main Jaina works on mathematics date from around 300BC to 400AD, but the Jaina religion was in its infancy as far back as 500BC. Further, as a point of slight interest, the Jaina religion has still not completely died out, and up till the 19th century some very minor works were produced.
Jaina mathematics played an important role in bridging the gap between 'ancient' Indian mathematics and the so-called 'Classical period', which was heralded by the work of Aryabhata I in the 6th century.
Regrettably there are few extant Jaina works, but in the limited material that exists an incredible level of originality is demonstrated. Perhaps the most historically important Jaina contribution to mathematics as a subject is the progression of the subject from purely practical or religious requirements. During the Jaina period mathematics became an abstract discipline to be cultivated "for its own sake". G Joseph states:
...The Jaina contribution to this change should be recognised. [GJ, P 250]
Despite its important historical position, relatively little attention has been paid to Jaina mathematics, and it remains seemingly discarded by many historians. This however is a massive oversight, because within Jaina works there are many remarkable 'new' results, and developments of topics found in Vedic works. It is worth briefly noting here that there is great uncertainty as to whether Vedic works influenced the Jaina mathematicians. Throughout the history of Indian mathematics it seems very possible each 'leap' was made without knowledge of previous discoveries. This has been suggested to be primarily due to the size of the sub-continent and imperfect channels of communication within it. Another contribution was the 'oral transmission' tradition of Ancient Indian mathematics, which resulted in knowledge being lost over time.
There are several significant Jaina works, including the Surya Prajinapti (4th c. BC) and several Sutras. There is also evidence of individual mathematicians including Bhadrabahu (c. 300 BC, possibly lived in the Mysore State), C Srinivasiengar conjectures he wrote two works (which have yet to be unearthed), and Umaswati, a commentator from 2nd century BC, (possibly lived around 150 BC). Umaswati is known as a great writer on Jaina metaphysics but also wrote a work Tattwarthadhigama-Sutra Bhashya, which contains mathematics.
Among the important developments of the Jainas are:
Theory of numbers.
There is a great fascination in Jain philosophy with the enumeration of large numbers, selected examples of 'time periods' mentioned include 756 1011 8400000028 days, called shirsa prahelika, and 2588 years.
All numbers were classified into three sets:
Enumerable, Innumerable and Infinite.
Five different types of infinity are recognised in Jaina works:
Infinite in one and two directions, infinite in area, infinite everywhere and infinite perpetually.
This theory is quite incredible and was not realised in Europe until the late 19th century work of George Cantor. Indeed much of the Jaina theory of infinity is extremely advanced for the time in which it was conceived.
Also found in Jaina works:
Knowledge of the fundamental laws of indices.
Operations with fractions.
Quartic equations (the Jaina contribution to algebra is severely neglected).
Formula for p (root 10, comes up almost inadvertently in a problem about infinity).
Operations with logarithms.
Sequences and progressions.
Finally of interest is the appearance of Permutations and Combinations in Jaina works, which resulted in the formation of an early Pascal triangle, called Meru Prastara, many centuries before Pascal himself 'invented' it. This is another case where Indian contributions have been neglected severely.
Example 5.1: Meru Prastara rule found in Jain works.
Rule is simpler than that of Pascal, and is based on the simple formula:
n+1Cr = nCr + nCr-1
Example 5.2: Formulas for permutations and combinations.
Correct formulas for both permutations and combinations are found in Jaina works:
nC1 = n, nC2 = n(n - 1)/1.2, nC3 = n(n - 1)(n - 2)/1.2.3
nP1 = n, nP2 = n(n - 1), nP3 = n(n - 1)(n - 2)
The contribution of the Buddhist school should also be briefly discussed. Although in the shadow of Jaina developments, evidence suggests Buddhist scholars were well versed in the use of the decimal place values system and that knowledge of Gainta was considered important.
There was no sudden decline of the Jaina religion as such but from the beginning of the 6th century the work of a mathematician named Aryabhata surpassed all previous work of the Indian sub-continent and brought about the 'Classical period' of Indian mathematics, (which lasted 600 years). However prior to discussing the work of Aryabhata there is a major piece of Indian mathematical work yet to be discussed. That is the Bakhshali manuscript.
(4 II. Sulba Sutras)
(6: The Bakhshali manuscript)