# Indian Mathematics - Redressing the balance

### Ian G Pearce

### The Classical period: VII. The end of the Classic period

The work of Bhaskara was considered the highest point Indian mathematics attained, and it was long considered that Indian mathematics ceased after that point. Extreme political turmoil through much of the sub-continent shattered the atmosphere of discovery and learning and led to the stagnation of mathematical developments as scholars contented themselves with duplicating earlier works.

Recent discoveries however have found that, despite political turmoil, mathematics continued to a high degree in the south of India up to the 16

Before going on to discuss the Kerala contribution to mathematics it is worth noting that by the time of Bhaskara II's death Indian mathematics of the 5

Sadly there is very little recognition of these facts, and even though the Arabic (and hence some Indian) works were prevalent in Spain, they did not transmit any further into Europe, which was still to fully 'awaken and probably 'resisted' the works, and many were subsequently lost. However ultimately a few Latin translations of Indo-Arabic works did flow into wider Europe, causing a step towards the renaissance.

This brief return to this discussion is primarily to serve as a reminder that Indian mathematics has had a far greater influence on the forward progress of mathematics (in conjunction with enlightened Arab scholars, and ultimately a handful of pioneering European scholars) than is generally mentioned. A prime example is the work of Fibonacci, which shows appreciation of Indo-Arabic work as early as the 12th century. In short, around the time of Bhaskara's death (12th c.) Indian mathematics (of the 5

As mentioned, it was long considered that following the 'high point' of the work of Bhaskara that Indian mathematics fell into a steep decline. To some extent this is true, only shortly after Bhaskara's death India was engulfed in war (Mongol invasions) and political turmoil. Consequently the atmosphere of security and tranquillity was lost, which was doubtlessly a major contributory factor to the barrenness of scientific activity and achievement. As C Srinivasiengar quotes:

There were occasional small developments, and attempts to revive learning, but nothing of the magnitude of the previous millennium.

Worthy of a brief mention are both Kamalakara (c 1616-1700) and Jagannatha Samrat (c. 1690-1750). Both combined traditional ideas of Indian astronomy with Arabic (and some Greek) concepts, Kamalakara gave trigonometric results of interest and Samrat made several Sanskrit translations of Arabic 'versions' of Greek works, including notably Euclid's Elements. However, as C Srinivasiengar comments:

Indeed his work contained new proofs not given by Euclid.

Under the patronage of monarch Sawai Jayasinha Raja (and his predecessor) Samrat was attempting, along with a group of scholars, to 'reinvigorate' science and learning in India. It must be admitted that these efforts do not appear to have been wholly successful, although the efforts were in the 'greatest of faith' and should be 'applauded'. To some extent the work of these two scholars only serves to further highlight the lack of originality and indolent attitude that was present by this stage in the north of India, although their efforts should not be completely ignored.

It must be considered most unfortunate that a country, which on reflection was unarguably a world leader in the field of mathematics for several thousands of years, ceased to contribute in any significant way. A theory has been suggested that, had there been definite 'links' between each 'major' period we have discussed, then India would have led the world, unequivocally, in the field of mathematics and may have continued to for much longer. However discoveries that have been made in the last 150 years have significantly altered the chronology of Indian mathematics and the way in which we should view Indian contributions.

Recent discoveries however have found that, despite political turmoil, mathematics continued to a high degree in the south of India up to the 16

^{th}century. The South of India avoided the worst of the political upheavals of the subcontinent, and the Kerala School of mathematics flourished for some time, producing some truly remarkable results. These results, the most notable of which are in the field of infinite series expansions of trigonometric functions, are generally inaccurately attributed to great European mathematicians of the 18^{th}century including Newton, Leibniz and Gregory. However, slowly, this rigid position is shifting somewhat.Before going on to discuss the Kerala contribution to mathematics it is worth noting that by the time of Bhaskara II's death Indian mathematics of the 5

^{th}and 6^{th}centuries had exerted a significant influence on mathematics across the world. By the 11^{th}century a number of important Arabic works had been written, based on translations of a number of Indian astronomical works. As the Arab Empire stretched as far as southern Spain, much of this work based on Indian science made its way into southern Europe and was subsequently translated into Latin.Sadly there is very little recognition of these facts, and even though the Arabic (and hence some Indian) works were prevalent in Spain, they did not transmit any further into Europe, which was still to fully 'awaken and probably 'resisted' the works, and many were subsequently lost. However ultimately a few Latin translations of Indo-Arabic works did flow into wider Europe, causing a step towards the renaissance.

This brief return to this discussion is primarily to serve as a reminder that Indian mathematics has had a far greater influence on the forward progress of mathematics (in conjunction with enlightened Arab scholars, and ultimately a handful of pioneering European scholars) than is generally mentioned. A prime example is the work of Fibonacci, which shows appreciation of Indo-Arabic work as early as the 12th century. In short, around the time of Bhaskara's death (12th c.) Indian mathematics (of the 5

^{th}-7^{th}centuries) was still exerting a significant influence throughout the world.As mentioned, it was long considered that following the 'high point' of the work of Bhaskara that Indian mathematics fell into a steep decline. To some extent this is true, only shortly after Bhaskara's death India was engulfed in war (Mongol invasions) and political turmoil. Consequently the atmosphere of security and tranquillity was lost, which was doubtlessly a major contributory factor to the barrenness of scientific activity and achievement. As C Srinivasiengar quotes:

...India suddenly fell into a state of "torpor" and never recovered from this torpor until the advent of mathematicians trained according to Western methods.[CS, P 142]

There were occasional small developments, and attempts to revive learning, but nothing of the magnitude of the previous millennium.

Worthy of a brief mention are both Kamalakara (c 1616-1700) and Jagannatha Samrat (c. 1690-1750). Both combined traditional ideas of Indian astronomy with Arabic (and some Greek) concepts, Kamalakara gave trigonometric results of interest and Samrat made several Sanskrit translations of Arabic 'versions' of Greek works, including notably Euclid's Elements. However, as C Srinivasiengar comments:

...Jagannatha's work was not mere translation.[CS, P 143]

Indeed his work contained new proofs not given by Euclid.

Under the patronage of monarch Sawai Jayasinha Raja (and his predecessor) Samrat was attempting, along with a group of scholars, to 'reinvigorate' science and learning in India. It must be admitted that these efforts do not appear to have been wholly successful, although the efforts were in the 'greatest of faith' and should be 'applauded'. To some extent the work of these two scholars only serves to further highlight the lack of originality and indolent attitude that was present by this stage in the north of India, although their efforts should not be completely ignored.

It must be considered most unfortunate that a country, which on reflection was unarguably a world leader in the field of mathematics for several thousands of years, ceased to contribute in any significant way. A theory has been suggested that, had there been definite 'links' between each 'major' period we have discussed, then India would have led the world, unequivocally, in the field of mathematics and may have continued to for much longer. However discoveries that have been made in the last 150 years have significantly altered the chronology of Indian mathematics and the way in which we should view Indian contributions.