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== The discovery of the Kerala School of Mathematics ==

== The discovery of the Kerala School of Mathematics ==

A relatively recently discovered field is what goes by the name of the “Kerala School of Mathematics” which flourished in a tiny corner of present-day Kerala during ~ 1300-1600 CE.  Many details about the work of this school and the story of the mathematicians who contributed to it are only now being researched. This despite the fact that this work was brought to the attention of western scientists almost 200 years ago.  In 1834,  an Englishman named Charles  M. Whish  published an article  entitled “On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four sastras, the Tantrasangraham, Yukti-Bhasha, Caruna-Padhati and Sadratnamala” in a journal called the ‘Transactions of the Royal Asiatic Society’ of Great Britain and Ireland. But the article was long ignored.

A relatively recently discovered field is what goes by the name of the “Kerala School of Mathematics” which flourished in a tiny corner of present-'''day Kerala during ~ 1300-1600 CE'''.  Many details about the work of this school and the story of the mathematicians who contributed to it are only now being researched. '''This despite the fact that this work was brought to the attention of western scientists almost 200 years ago'''.  In 1834,  an Englishman named Charles  M. Whish  published an article  entitled <blockquote>“On the Hindu quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four sastras, the Tantrasangraham, Yukti-Bhasha, Caruna-Padhati and Sadratnamala” in a journal called the ‘Transactions of the Royal Asiatic Society’ of Great Britain and Ireland. '''But the article was long ignored'''.

  What was the main contribution of the Kerala school?

 The Kerala school of mathematicians drew inspiration from much earlier texts, mainly Āryabhata’s Āryabhatiya (499 CE). The Āryabhatiya had in fact been a very influential text all over the country, and also, through its translations, in the Arab world and in Europe.

 

 The Kerala mathematicians, starting with Mādhava, developed some amazing mathematics – in particular, the branch of mathematics that is known today as Calculus, one of the foundation stones of modern science which developed from Europe.

== The main contribution of the Kerala school ==

 We have all been taught in school that Calculus was discovered by (Leibniz and) Newton. But Newton’s Magnum Opus, the Principia Mathematica, in which he discusses the Calculus essential for his Laws of Motion, was written around ~1700 CE. Thus, even orthodox historians and scientists now agree that the Kerala Calculus pre-dates that of Newton by at least a clear 200 years.

'''The Kerala school of mathematicians drew inspiration from much earlier texts, mainly Āryabhata’s Āryabhatiya (499 CE). The Āryabhatiya had in fact been a very influential text all over the country, and also, through its translations, in the Arab world and in Europe.'''

   A little more on some of the contributions of the Kerala school

The Kerala mathematicians, starting with Mādhava, developed some amazing mathematics – in particular, the branch of mathematics that is known today as Calculus, one of the foundation stones of modern science which developed from Europe.

 

We have all been taught in school that Calculus was discovered by (Leibniz and) Newton. But Newton’s Magnum Opus, the Principia Mathematica, in which he discusses the Calculus essential for his Laws of Motion, was written around ~1700 CE. Thus, even orthodox historians and scientists now agree that the Kerala Calculus pre-dates that of Newton '''by at least a clear 200 years'''.

A little more on some of the contributions of the Kerala school

 Calculus is the mathematical study of change, and its essence is the use of  infinitesimals / limits  (and, one of the passages to “limit” is by summing an infinite series).

 Calculus is the mathematical study of change, and its essence is the use of  infinitesimals / limits  (and, one of the passages to “limit” is by summing an infinite series).

 The concept of limit as given by Nīlakantha in Āryabhatiya-bhāsya :

 The concept of limit as given by Nīlakantha in Āryabhatiya-bhāsya :

Instantaneous velocity (of planets) and derivatives.

Instantaneous velocity (of planets) and derivatives.

Besides arriving at the infinite series, that several forms of rapidly convergent series could be obtained is remarkable. Further, many equations that we use in Calculus which are attributed to western mathematicians were clearly known to the Indian mathematicians. They laid the foundations of Calculus, which is recognized as one of the foundations of modern science, and which has applications in many fields including engineering and economics.

Besides arriving at the infinite series, that several forms of rapidly convergent series could be obtained is remarkable. Further, many equations that we use in Calculus which are attributed to western mathematicians were clearly known to the Indian mathematicians. They laid the foundations of Calculus, which is recognized as one of the foundations of modern science, and which has applications in many fields including engineering and economics.

These mathematicians also made important contributions to astronomy, but those will be the subject of a separate article. In fact,  much of this work seems to have arisen from an interest in predicting planetary positions, sunrise, sunset etc. to a very high accuracy for the  conduct of worldly affairs.

These mathematicians also made important contributions to astronomy, but those will be the subject of a separate article. In fact,  much of this work seems to have arisen from an interest in predicting planetary positions, sunrise, sunset etc. to a very high accuracy for the  conduct of worldly affairs. 

 Who were these people ? – some historical details

 

'''Who were these people ? – some historical details'''

 

Most of these developments took place in temple-villages around a river called Nila in the ancient days (and currently called river Bharatha, the second longest river in Kerala) during ~ 1300-1600 CE. In fact, the area over which this work was carried out was so localized, that some scholars suggest that the school is more appropriately named the “Nila School of Mathematics”. One of the key villages was Sangamagrāma, which was possibly the present-day village of Irinhalakkuta (about 50 km to the south of Nila). (However, there are a few other possible candidates for Sangamagrāma , such as Kudalur and Tirunavaya). What is more certain was the existence of a remarkable lineage of mathematicians in and around Sangama-grama of which the pioneer,  Mādhava (~1340-1420)   seems to be the one who discovered many of the basic ideas of Calculus.

Most of these developments took place in temple-villages around a river called Nila in the ancient days (and currently called river Bharatha, the second longest river in Kerala) during ~ 1300-1600 CE. In fact, the area over which this work was carried out was so localized, that some scholars suggest that the school is more appropriately named the “Nila School of Mathematics”. One of the key villages was Sangamagrāma, which was possibly the present-day village of Irinhalakkuta (about 50 km to the south of Nila). (However, there are a few other possible candidates for Sangamagrāma , such as Kudalur and Tirunavaya). What is more certain was the existence of a remarkable lineage of mathematicians in and around Sangama-grama of which the pioneer,  Mādhava (~1340-1420)   seems to be the one who discovered many of the basic ideas of Calculus.

The Kerala school was a culmination of the school of Āryabhata and seems to have been the last bastion of mathematics in India till the modern era. The school seems to have died out soon after the arrival of the Portuguese in Kerala for obvious historical reasons.

The Kerala school was a culmination of the school of Āryabhata and seems to have been the last bastion of mathematics in India till the modern era. '''The school seems to have died out soon after the arrival of the Portuguese in Kerala for obvious historical reasons. (Prof Ram)'''

== The Lineage ==

== The Lineage ==

Mādhava (c.1340–1420) of Sangamagrāma

Mādhava (c.1340–1420) of Sangamagrāma

 Pioneer of the Kerala School, discovered many of the basic ideas of Calculus.

 Pioneer of the Kerala School, discovered many of the basic ideas of Calculus.

 The only works of his which seem to be extant are Venvāroha and Sphutacandrāpati.

 

The only works of his which seem to be extant are Venvāroha and Sphutacandrāpati.

Parameśvara (c. 1380–1460) of Vatasseri

Parameśvara (c. 1380–1460) of Vatasseri

Mādhava’s disciple, great observer and prolific writer.

Mādhava’s disciple, great observer and prolific writer.