Kerala school of astronomy and mathematics

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The Kerala School was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala (South India), which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559-1632). In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c.1500-c.1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.^[1]

Their discovery of these three important series expansions of calculus—several centuries before calculus was developed in Europe by Leibniz and Newton—was a landmark achievement in mathematics. However, the Kerala School cannot be said to have invented calculus,^[2] because, while they were able to develop Taylor series expansions for the important trigonometric functions, they developed neither a comprehensive theory of differentiation or integration, nor the fundamental theorem of calculus.^[3]

1 Contributions
2 Prominent mathematicians
3 Possible transmission of Kerala mathematics to Europe
4 References
5 Bibliography
6 External Links
7 See also

[edit] Contributions

[edit] Infinite Series, and Calculus

The (infinite) geometric series: $\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots + \infty$ for $| x | < 1$ ^[4] This formula was already known, for example, in the work of the 10th century Arab mathematician Alhazen (the latinized form of the name Ibn Al-Haytham (965-1039)).^[5]
A semi-rigorous proof (see "induction" remark below) of the result: $1^p+ 2^p + \cdots + n^p \approx \frac{n^{p+1}}{p+1}$ for large n. This result was also known to Alhazen.^[1]
Intuitive use of mathematical induction, however, the inductive hypothesis was not formulated or employed in proofs.^[1]
Applications of ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for $sin x$ , $cos x$ , and $arctan x$ ^[2] The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:^[1]

$r\arctan(\frac{y}{x}) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^t} - \cdots ,$ where $y/x \leq 1.$

$\sin x = x - x\cdot\frac{x^2}{(2^2+2)r^2} + x\cdot \frac{x^2}{(2^2+2)r^2}\cdot\frac{x^2}{(4^2+4)r^2} - \cdot$

$r - \cos x = r\cdot \frac{x^2}{(2^2-2)r^2} - r\cdot \frac{x^2}{(2^2-2)r^2}\cdot \frac{x^2}{(4^2-4)r^2} + \cdots ,$ where, for

r = 1

, the series reduce to the standard power series for these trigonometric functions, for example:

- $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$ and
- $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$ (The Kerala school themselves did not use the "factorial" symbolism.)
Use of rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle, was not used.)^[1]
Use of series expansion of $arctan x$ to obtain an infinite series expression (later known as Gregory series) for $π$ :^[1]

$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots + \infty$

A rational approximation of error for the finite sum of their series of interest. For example, the error, $f i (n + 1)$ , (for n odd, and i = 1, 2, 3) for the series:

$\frac{\pi}{4} \approx 1 - \frac{1}{3}+ \frac{1}{5} - \cdots (-1)^{(n-1)/2}\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)$

where $f_1(n) = \frac{1}{2n}, \ f_2(n) = \frac{n/2}{n^2+1}, \ f_3(n) = \frac{(n/2)^2+1}{(n^2+5)n/2}.$

Manipulation of error term to derive a faster converging series for $π$ :^[1]

$\frac{\pi}{4} = \frac{3}{4} + \frac{1}{3^3-3} - \frac{1}{5^3-5} + \frac{1}{7^3-7} - \cdots \infty$

Using the improved series to derive a rational expression,^[1] $104348 / 33215$ for $π$ correct up to nine decimal places, i.e. $3.141592653$
Use of an intuitive notion of limit to compute these results.^[1]
A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions.^[3] However, they did not formulate the notion of a function, or have knowledge of the exponential or logarithmic functions.

The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries."^[6] However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers,^[7]^[8] a commentary on the Yuktibhasa's proof of the sine and cosine series^[9] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).^[10]^[11]

[edit] Geometry, Arithmetic, and Algebra

A formula for the ecliptic.^{[citation needed]}
Lhuilier's formula for the circumradius of a cyclic quadrilateral.^{[citation needed]}
Decimal floating point numbers.^{[citation needed]}
Iterative methods for solution of non-linear equations.^{[citation needed]}
The Newton-Gauss interpolation formula by Govindaswami.^{[citation needed]}

[edit] Astronomy and Linguistics

A procedure to determine the positions of the Moon every 36 minutes.^{[citation needed]}
Methods to estimate the motions of the planets.^{[citation needed]}
A formulation for the equation of the center of the planets.^{[citation needed]}
A heliocentric model of the solar system.^{[citation needed]}

The Kerala School also contributed much to linguistics:

The ayurvedic and poetic traditions of Kerala were founded by this school.^{[citation needed]}
The famous poem, Narayaneeyam, was composed by Narayana Bhattathiri.^{[citation needed]}

[edit] Prominent mathematicians

The neutrality and factual accuracy of this section are disputed.
Please see the relevant discussion on the talk page.

[edit] Madhava of Sangamagrama (1340-1425)

Madhava (c. 1340-1425) was the founder of the Kerala School. Although it is possible that he wrote Karana Paddhati a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars.

Little is known about Madhava, who lived near Kochi between the years 1340 and 1425. Nilkantha attributes the series for sine to Madhava. It is not known if Madhava discovered the other series as well, or whether they were discovered later by others in the Kerala school. Madhava's discoveries include:

Taylor series for the sine.^[3]
Second-order Taylor series approximations of the sine and cosine functions.^{[citation needed]}
Third-order Taylor series approximation of the sine function.^{[citation needed]}
Power series of π (usually attributed to Leibniz).
The solution of transcendental equations by iteration.^{[citation needed]}
Approximation of transcendental numbers by continued fractions.^{[citation needed]}
Correctly computed the value of $π$ to 9 decimal places.^[1]
Sine and cosine tables to 9 decimal places of accuracy.^{[citation needed]}

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

[edit] Narayanan Pandit (1340-1400)

Narayana Pandit, one among the notable Kerala mathematicians, had written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. Narayanan is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavathi, 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.

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

A rule to calculate approximate values of square roots.
Investigations into 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.

[edit] Parameshvaran (1370-1460)

Parameshvaran, the founder of the Drigganita system of Astronomy, was a prolific author of several important works. He belonged to the Alathur village situated on the bank of Bharathappuzha.He is stated to have made direct astronomical observations for fifty-five years before writing his famous work, Drigganita. He also wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavathi Bhasya, a commentary on Bhaskara II's Lilavathi, 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-Deepika by Paramesvara is a commentary on the commentary of Govindsvamin on Bhaskara I's Maha-bhaskareeya. It contains:

Some of his eclipse observations in this work including one made at Navakshethra in 1422 and two made at Gokarna in 1425 and 1430.
A mean value type formula for inverse interpolation of the sine function.
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 L'Huilier (1782).

[edit] Nilakanthan Somayaji (1444-1544)

Nilakantha was a disciple of Govinda, son of Parameshvara. He was a brahmin from Trkkantiyur in Ponnani taluk. His younger brother Sankara was also a scholar in astronomy. Nilakantha's most notable work Tantra Samgraha (which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yukthideepika, 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-bhashya a commentary of the Aryabhatiya. Of great significance in Nilakantha's work includes:

The presence of inductive mathematical proof.
Derivation and proof of the Madhava-Gregory series of the arctangent trigonometric function.
Improvements and proofs of other infinite series expansions by Madhava.
An improved series expansion of π/4 that converges more rapidly.
The relationship between the power series of π/4 and arctangent.
Sophisticated explanations of the irrationality of π.
The correct formulation for the equation of the center of the planets.
A true heliocentric model of the solar system.

[edit] Chitrabhanu (circa 1530)

Citrabhanu was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous Diophantine 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, Chitrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.

[edit] Jyesthadevan (circa 1500-1600)

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

Proofs of theorems.
Derivations of rules and series.
Derivation and proof of the Madhava-Gregory series of the arctangent function.
Proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other mathematicians of the Kerala School.
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 kuttakaranam method.
Rules of finding the sines and the cosines of the sum and difference of two angles.

Jyesthadevan also gave:

The earliest statement of Wallis' theorem.
Geometrical derivations of series.

[edit] Sankaran Varma (1800-1838)

There remains a final Kerala work worthy of a brief mention, Sadratnamala an astronomical treatise written by Sankara Varman that 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 out as the last notable name in Keralan mathematics. A remarkable contribution was his compution of π correct to 17 decimal places.

[edit] Possible transmission of Kerala mathematics to Europe

The neutrality and factual accuracy of this section are disputed.
Please see the relevant discussion on the talk page.

There are a number of publications, including a recent paper of interest written by D. Almeida, J. John and A. Zadorozhnyy, which suggest Keralan 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 direct 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 the Pythagorean 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 Islamic scholars (see Bhaskara: Influence) or through Kerala 'routes'.

Although it was believed that Kerala calculus remained localised until its discovery by Charles Whish in 1832, Kerala had 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 Kerala mathematics. Indian mathematical manuscripts may have been brought to Europe by the Jesuit priests and scholars that were present in Kerala.

In particular, it is well-known that Matteo Ricci, the Jesuit mathematician and astronomer who is generally credited with bringing European science and mathematics to China, spent two years in Kochi, Kerala after being ordained in Goa in 1580. During that time he was in correspondence with the Rector of the Collegio Romano, the primary institution for the education of those who wished to become Jesuits. Matteo Ricci wrote back to Petri Maffei stating that he was seeking to learn the methods of timekeeping from "an intelligent Brahman or an honest Moor". The Jesuits at the time were very knowledgeable in science and mathematics, and many were trained as mathematicians at the Jesuit seminaries. For a number of Jesuits who followed Ricci, Kochi was a staging point on the way to China. Kochi was only 70km away from the largest repository of Kerala's mathematical and 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 Padua, where James Gregory studied; at Paris, where Marin Mersenne, through his correspondence with Pierre de Fermat, Blaise Pascal, Galileo and Wallis, acted as an agent for the transmission of mathematical ideas. It is quite possible that these mathematical ideas transmitted by the Jesuits included mathematics from Kerala.

Other pieces of circumstantial evidence include:

James Gregory, who first stated the infinite series expansion of the arctangent (the Madhava-Gregory series) in Europe, never gave any derivation of his result, or any indication as to how he derived it, suggesting that this series was imported into Europe. (See Infinitesimal Calculus - How and why it was imported to Europe.)
Kerala's established 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.

[edit] References

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j Roy, Ranjan. 1990. "Discovery of the Series Formula for $π$ by Leibniz, Gregory, and Nilakantha." Mathematics Magazine (Mathematical Association of America) 63(5):291-306.
^ ^a ^b Bressoud, David. 2002. "Was Calculus Invented in India?" The College Mathematics Journal (Mathematical Association of America). 33(1):2-13.
^ ^a ^b ^c Katz, V. J. 1995. "Ideas of Calculus in Islam and India." Mathematics Magazine (Mathematical Association of America), 68(3):163-174.
^ Singh, A. N. Singh. 1936. "On the Use of Series in Hindu Mathematics." Osiris 1:606-628.
^ Edwards, C. H., Jr. 1979. The Historical Development of the Calculus. New York: Springer-Verlag.
^ Charles Whish (1835). Transactions of the Royal Asiatic Society of Great Britain and Ireland.
^ Rajagopal, C. and M. S. Rangachari. 1949. "A Neglected Chapter of Hindu Mathematics." Scripta Mathematica. 15:201-209.
^ Rajagopal, C. and M. S. Rangachari. 1951. "On the Hindu proof of Gregory's series." Ibid. 17:65-74.
^ Rajagopal, C. and A. Venkataraman. 1949. "The sine and cosine power series in Hindu mathematics." Journal of the Royal Asiatic Society of Bengal (Science). 15:1-13.
^ Rajagopal, C. and M. S. Rangachari. 1977. "On an untapped source of medieval Keralese mathematics." Archive for the History of Exact Sciences. 18:89-102.
^ Rajagopal, C. and M. S. Rangachari. 1986. "On Medieval Kerala Mathematics." Archive for the History of Exact Sciences. 35:91-99.

[edit] Bibliography

Hayashi, Takao. 1997. "Number Theory in India". In Helaine Selin, ed. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Boston: Kluwer Academic Publishers, pp. 784-786
K V Sarma, and S Hariharan: Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy - an analytical appraisal, Indian J. Hist. Sci. 26 (2) (1991), 185-207
Plofker, Kim, ‘An example of the secant method of iterative approximation in a fifteenth-century Sanscrit text’, Historia mathematica 23 (1996), 246-256
Parameswaran, S., ‘Whish’s showroom revisited’, Mathematical gazette 76, no. 475 (1992) 28-36
R G Gupta,"Second Order of Interpolation of Indian Mathematics", Ind, J.of Hist. of Sc. 4 (1969) 92-94
George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition, Penguin Books, 2000.
Victor J. Katz. A History of Mathematics: An Introduction, 2nd Edition, Addison-Wesley, 1998.
T. R. N. Rao and Subhash C. Kak. Computing Science in Ancient India, USL Press, Lafayette, 1998.
C. K. Raju. 'Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhâsâ', Philosophy East and West 51, University of Hawaii Press, 2001.
Tacchi Venturi. 'Letter by Matteo Ricci to Petri Maffei on 1 Dec 1581', Matteo Ricci S.I., Le Lettre Dalla Cina 1580–1610, vol. 2, Macerata, 1613.

[edit] External Links

The Kerala School, European Mathematics and Navigation, 2001.
An overview of Indian mathematics, MacTutor History of Mathematics archive, 2002.
Indian Mathematics: Redressing the balance, MacTutor History of Mathematics archive, 2002.
Keralese mathematics, MacTutor History of Mathematics archive, 2002.
Possible transmission of Keralese mathematics to Europe, MacTutor History of Mathematics archive, 2002.
Neither Newton nor Leibnitz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala, 2005.