Kerala school of astronomy and mathematics

The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and has its intellectual roots with Aryabhatta who lived in the 5th century. The lineage continues down to modern times but the original research seems to have ended with Narayana Bhattathiri (1559-1632). These astronomers, in attempting to solve problems, invented a number of ideas including:

• Revolutionary ideas of calculus.
• The theory of infinite series.
• Infinite series expansions of functions.
• Power series.
• Taylor series.
• Trigonometric series.
• Tests of convergence (often attributed to Cauchy).
• Decimal floating point numbers
• Using the floating point system of numbers, they were able to investigate and rationalise about the convergence of series.
• The formula for the sum of an infinite series.
• Methods of differentiation.
• Integration.
• Term by term integration.
• Numerical integration by means of infinite series.
• The theory that the area under a curve is its integral.
• Used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.
• Iterative methods for solution of non-linear equations.
• The Newton-Gauss interpolation formula.
• A formula for the ecliptic.
• Lhuilier's formula for the circumradius of a cyclic quadrilateral.
• The first approximation to the value of π made using a series.

They achieved most of these results centuries before European mathematicians. Jyeshtadeva consolidated the Kerala School's discoveries in the Yuktibhasa, the world's first calculus text. In many ways, the Kerala School represents the peak of mathematical knowledge in the middle ages.

According to Charles Whish in 1835, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works were "abound with fluxional forms and series to be found in no work of foreign countries."

The Kerala School also contributed much to linguistics. The ayurvedic and poetic traditions of Kerala were founded by this school. The famous poem, Narayaneeyam, was composed by Narayana Bhattathiri.

Keralese Mathematicians

Narayana Pandit (1340-1400)

Narayana Pandit, the earliest of the notable Kerala mathematicians, had written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled Karmapradipika (or Karma-Paddhati).

Although the Karmapradipika contains little original work, the following are found within it:

• Seven different methods for squaring numbers, a contribution that is wholly original to the author.
• Contributions to algebra.
• Contributions to magic squares.

Narayana's other major works contain a variety of mathematical developments, including:

• A rule to calculate approximate values of square roots.
• The second order indeterminate equation nq² + 1 = p² (Pell's equation).
• Solutions of indeterminate higher-order equations.
• Mathematical operations with zero.
• Several geometrical rules.
• Discussion of magic squares and similar figures.
• Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work.
• Narayana has also made contributions to the topic of cyclic quadrilaterals.

Madhava of Sangamagramma (1340-1425)

Madhava of Sangamagramma was the founder of the Kerala School and considered to be one of the greatest mathematician-astronomers of the Middle Ages. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475 but all we really know of Madhava comes from works of later scholars.

Perhaps his most significant contribution was in:

• Moving on from the finite procedures of ancient mathematics to treat their limit passage to infinity, which is considered to be the essence of modern classical analysis, and thus he is considered the father of mathematical analysis.

Madhava was also responsible for many other significant and original discoveries, including:

• Infinite series expansions of functions.
• The power series.
• The Taylor series.
• Trigonometric series.
• Rational approximations of infinite series.
• Taylor series of the sine and cosine functions (Madhava-Newton power series).
• Taylor series of the tangent function.
• Taylor series of the arctangent function (Madhava-Gregory series).
• Second-order Taylor series approximations of the sine and cosine functions.
• Third-order Taylor series approximation of the sine function.
• Power series of π (usually attributed to Leibniz).
• Power series of π/4 (Euler's series).
• Power series of the radius.
• Power series of the diameter.
• Power series of the circumference.
• Power series of angle θ (equivalent to the Gregory series).
• Infinite continued fractions.
• Integration.
• Term by term integration.
• The solution of transcendental equations by iteration.
• Approximation of transcendental numbers by continued fractions.
• Tests of convergence of infinite series.
• Correctly computed the value of π to 11 decimal places, the most accurate value of π after almost a thousand years.
• Sine and cosine tables to 9 decimal places of accuracy, which would remain the most accurate upto the 17th century.
• Laying the foundations for the development of calculus, which was then further developed by his successors at the Kerala School.

He also extended some results found in earlier works, including those of Bhaskara.

Parameshvara (1370-1460)

Parameshvara wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his most important discoveries:

• An outstanding version of the mean value theorem, which is the most important result in differential calculus and one of the most important theorems in mathematical analysis. This result was later essential in proving the fundamental theorem of calculus.

The Siddhanta-dipika by Paramesvara is a commentary on the commentary of Govindsvamin on Bhaskara I's Maha-bhaskariya. It contains:

• Some of his eclipse observations in this work including one made at Navaksetra in 1422 and two made at Gokarna in 1425 and 1430.
• A mean value type formula for inverse interpolation of the sine.
• It presents a one-point iterative technique for calculating the sine of a given angle.
• A more efficient approximation that works using a two-point iterative algorithm, which is essentially the same as the modern secant method.

He was also the first mathematician to:

• Give the radius of circle with inscribed cyclic quadrilateral, an expression that is normally attributed to Lhuilier (1782).

Nilakantha Somayaji (1444-1544)

In Nilakantha's most notable work Tantra Samgraha (which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yuktidipaika, written in 1501) he elaborates and extends the contributions of Madhava. Sadly none of his mathematical works are extant, however it can be determined that he was a mathematician of some note. Nilakantha was also the author of Aryabhatiya-bhasa a commentary of the Aryabhatiya. Of great significance in Nilakantha's work includes:

• The presence of inductive mathematical proof.
• Proof of the Madhava-Gregory series of the arctangent.
• Improvements and proofs of other infinite series expansions by Madhava.
• An imporved series expansion of π/4 that converges more rapidly.
• The relationship between the power series of π/4 and arctangent.

Citrabhanu (c. 1530)

Citrabhanu was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:

$\ x+y=a,x-y=b,xy=c,x^{2}+y^{2}=d,x^{2}-y^{2}=e,x^{3}+y^{3}=f,x^{3}-y^{3}=g$

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.

Jyesthadeva (1500-1575)

Jyesthadeva was another member of the Kerala School. His key work was the Yukti-bhasa (written in Malayalam, a regional language of Kerala), the world's first calculus text. It contained most of the developments of earlier Kerala School mathematicians, particularly Madhava. Similarly to the work of Nilakantha, it is almost unique in the history of Indian mathematics, in that it contains:

• Proofs of theorems.
• Derivations of rules and series.
• Proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians.
• Proof of the series expansion of the arctangent function (equivalent to Gregory's proof), and the sine and cosine functions.

He also studied various topics found in many previous Indian works, including:

• Integer solutions of systems of first degree equations solved using kuttaka.
• Rules of finding the sines and the cosines of the sum and difference of two angles.

Jyesthadeva also gave:

• The earliest statement of Wallis' theorem.
• Geometric derivations of series.

Sankara Varman (1800-1838)

There remains a final Kerala work worthy of a brief mention, Sadrhana-Mala an astronomical treatise written by Sankara Varman serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19th century and the author stands as the last notable name in Keralese mathematics. A notable contribution was his compution of π correct to 17 decimal places.

Possible transmission of Keralese mathematics to Europe

There are a number of publications, including a recent paper of great interest written by D. Almeida, J. John and A. Zadorozhnyy, which suggest Keralese mathematics may have been transmitted to Europe. Kerala was in continuous contact with China, Arabia, and from around 1500, Europe as well, thus transmission would have been possible. There is no evidence by way of relevant manuscripts but the evidence of methodological similarities, communication routes and a suitable chronology for transmission is hard to dismiss.

A key development of pre-calculus Europe, that of generalisation on the basis of induction, has deep methodological similarities with the corresponding Kerala development (200 years before). There is further evidence that John Wallis (1665) gave a recurrence relation and proof of Pythagoras theorem exactly as Bhaskara II did. The only way European scholars at this time could have been aware of the work of Bhaskara would have been through Keralese 'routes'.

Although it was believed that Keralese calculus remained localised until its discovery by Charles Whish in 1832, Kerala has in fact been in contact with Europe ever since Vasco da Gama first arrived there in 1499 and trade routes were established between Kerala and Europe. Along with European traders, Jesuit missionaries from Europe were also present in Kerala during the 16th century. Many of them were mathematicians and astronomers, and were able to speak local languages such as Malayalam, and were thus able to comprehend Keralese mathematics. Indian mathematical manuscripts may have been brought to Europe by the Jesuit priests and scholars that were present in Kerala.

In particular, there is strong evidence that Matteo Ricci, the Jesuit mathematician and astronomer who is generally credited with bringing European science and mathematics to China, spent two years in Cochin, Kerala after being ordained in Goa in 1580. During that time he was in correspondance with the Rector of the Collegio Romano, the primary institution for the education of those who wished to become Jesuits. The Jesuits at the time were very knowledgable in science and mathematics, and many were trained as mathematicians at the Jesuit seminaries. For a number of Jesuits who followed Ricci, Cochin was a staging point on the way to China. Cochin (now known as Kochi) was only 70km away from the largest repository of astronomical documents in Thrissur (Trichur). This was where, 200 years later, the European mathematicians Charles Whish and Heyne obtained their copies of manuscripts written by the Kerala mathematicians.

The Jesuits were expected to regularly submit reports to their headquarters in Rome, and it is possible that some of the reports may have contained appendices of a technical nature which would then be passed on by Rome to those who understood them, including notable mathematicians. Material gathered by the Jesuits was scattered all over Europe: at Pisa, where Galileo Galilei, Bonaventura Cavalieri and [[John Wallis|]] spent time; at Padau, where James Gregory studied; at [[Paris], where Marin Mersenne, through his correspondance with Pierre de Fermat, Blaise Pascal, Galileo and Wallis, acted as an agent for the transmission of mathematical ideas. It is possible that mathematical ideas from Kerala was also spread by the Jesuits.

Other pieces of evidence include Kerala's trade links with the British East India Company, which began trading with India sometime between 1600 and 1608, not too long before Europe's Scientific Revolution began. There was also some controversy in the late 17th century between Newton and Leibniz, over how they independently 'invented' calculus almost simultaneously, which sometimes leads to the suggestion that they both may have acquired the relevant ideas indirectly from Keralese calculus.

Sources

• George G. Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition, Penguin Books, 2000.
• D. P. Agrawal. The Kerala School, European Mathematics and Navigation, 2001.
• Dr. John J. O'Connor and Professor Edmund F. Robertson. 'An overview of Indian mathematics', MacTutor History of Mathematics archive, St Andrews University, 2002.
• Ian G. Pearce. 'Indian Mathematics: Redressing the balance', MacTutor History of Mathematics archive, 2002.
• Ian G. Pearce. 'Keralese mathematics', MacTutor History of Mathematics archive, 2002.
• Ian G. Pearce. 'Possible transmission of Keralese mathematics to Europe', MacTutor History of Mathematics archive, 2002.
• Dr. Sarada Rajeev. Neither Newton nor Leibnitz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala, University of Rochester, 2005.

See also

• Indian mathematics
• Indian mathematicians
• History of mathematics