Astronomy were provided a century later in

Astronomy were provided a century later in

Astronomy (Jyothissaasthram) was popular in Kerala even in ancient times, and their deep knowledge in that branch of science is well-known. A number of great treatises (Grantham) were written by several eminent scholars (most of them Namboothiri Brahmins) of the area at different times.

It is difficult to date some of the very ancient ones such as “Devakeralam”, “Sukrakeralam” (also known as “Bhrigukeralam”, “Kerala Rahasyam” or “Keraleeyam” and has 10 chapters), “Vararuchi Keralam” and “Keraleeya Soothram”. Jyothissaasthram was divided in to three Skandhhams (branches) – Ganitham, Samhitha and Hora.While we consider the contributions to astronomy and mathematics from Kerala in the pre modern era ,the role of Kerala school of astronomy and mathematics is highly remarkable.

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Kerala school of astronomy and mathematics The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Malabar, 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, calledTantrasangraha-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. Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.Contributions Infinite Series and Calculus The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following (infinite) geometric series:  for | x | < 1 This formula, however, was already known in the work of the 10th century  Iraqi  mathematician  Alhazen (the Latinized form of the name Ibn al-Haytham) (965–1039). The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs.

They used this to discover a semi-rigorous proof of the result:  for large n. This result was also known to Alhazen. They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for sinx, cosx, andarctanx .

The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:  where where, for r = 1, the series reduce to the standard power series for these trigonometric functions, for example:  and The Kerala school themselves did not use the “factorial” symbolism. ) The Kerala school made use of the 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 yet developed. ) They also made use of the series expansion of arctanx to obtain an infinite series expression (later known as Gregory series) for ? : Their rational approximation of the error for the finite sum of their series are of particular interest.For example, the error, fi(n + 1), (for nodd, and i = 1, 2, 3) for the series: Where they manipulated the error term to derive a faster converging series for ? They used the improved series to derive a rational expression, 104348 / 33215 for ? correct up to nine decimal places, i.

e. 3. 141592653. They made use of an intuitive notion of a limit to compute these results.

The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions, though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835, though there exists some other works, namely Kala Sankalita by J. Warren in 1825  which briefly mentions the discovery of infinite series by Kerala astronomers. 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.

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, a commentary on the Yuktibhasa’s proof of the sine and cosine series and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary). Geometry, Arithmetic, and AlgebraIn the fields of geometry, arithmetic, and algebra, the Kerala school discovered a formula for the ecliptic, Lhuilier’s formula for the circumradius of a cyclic quadrilateral by Parameshvara, decimal floating point numbers, the secant method and iterative methods for solution of non-linear equations by Parameshvara, and the Newton-Gauss interpolation formula by Govindaswami. Astronomy In astronomy, Madhava discovered a procedure to determine the positions of the Moon every 36 minutes, and methods to estimate the motions of the planets.Late Kerala school astronomers gave a formulation for the equation of the center of the planets, and a heliocentric model of the solar system. In 1500, Nilakanthan Somayaji (1444–1544) of the Kerala school of astronomy and mathematics, in his Tantrasangraha, revised Aryabhata’s model for the planets Mercury and Venus.

His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century.Nilakanthan Somayaji, in his ‘Aryabhatiyabhasya’, a commentary on Aryabhata’s ‘Aryabhatiya’, developed his own computational system for a partially heliocentric planetary model, in which Mercury, Venus, Mars, Jupiter and Saturn orbit the Sun, which in turn orbits the Earth, similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century. Nilakantha’s system, however, was mathematically more efficient than the Tychonic system, due to correctly taking into account the equation of the centre and latitudinal motion of Mercury and Venus.Most astronomers of the Kerala school of astronomy and mathematics who followed him accepted his planetary model. Prominent mathematicians Madhavan of Sangamagrama (1340 – 1425) Madhavan of Sangamagraamam, as he is known, holds a position of eminence among the astute astronomers of medieval Kerala. He hailed from Sangama Graamam, the modern Irinjalakuda, near the railway station. Madhavan was the teacher of Parameswaran, the promulgator of Drigganitha school of Astronomy, and is frequently quoted in the medieval astronomical literature of Kerala as Golavith (adept in spherics).

Little is known about Madhava, who lived at Irinjalakuda,at that time known as Iringattikudal in Thrissur district between the years 1340 and 1425. Sanskrit scholars used to call the town as Sangamagramam, taking into consideration of the meaning of Kudal apprearing in Iringattikudal, which has the meaning Sangamam in Sanskrit. Nilkantha attributes the series for sine to him. It is not known if Madhava discovered the other series as well, or whether they were discovered later by others in the Kerala school He is the author of several important treatises on Mathematics and Astronomy.

The “Venvaaroham” explaining the method for computation of the moon and the moon-sentences, “Aganitham”, an extensive treatise on the computation of planets, “Golavaadam”, “Sphhuta-Chandraapthi”, “Madhyama Nayana Prakaaram” are some of his important works. Besides these works, a number of stray verses of Madhavan are quoted by later astronomers like Neelakandha Somayaaji, Narayanan the commentator of Leelaavathy, Sankaran the commentator of Thanthrasangraham, etc.One of his significant contributions is his enunciatiation of formulae for accurate determination of the circumference of a circle and the value of ? by the method of indeterminate series, a method which was rediscovered in Europe nearly three centuries later by James Gregory (1638 – 75 AD), Gottfried Wilhelm Leibniz (1646 – 1716 AD) and Newton (1642, “Principia Mathematicia”). His five Paraspara-Nyaaya contains the enunciation for the first time in the world, of the formula for the sine of sum of two angles. sin(A+B) = sinAcosB+cosAsinBThis is known as “Jeeve Paraspara Nyaaya”. Madhava’s discoveries include the Taylor series for the sine, cosine, tangent and arctangent functions; the second-order Taylor series approximations of the sine and cosine functions and the third-order Taylor series approximation of the sine function; the power series of ? , usually attributed to Leibniz but now known as the Madhava-Leibniz series; the solution of transcendental equations byiteration; and the approximation of transcendental numbers by continued fractions.

Madhava correctly computed the value of? to 9 decimal places and 13 decimal places, and produced sine and cosine tables to 9 decimal places of accuracy. He also extended some results found in earlier works, including those of Bhaskara. The ideas of Calculus and Trigonometry were developed by him in the middle of the 14th century itself, as can be verified by his extensive mathematical and astronomical treatises and quotations by later authors.

Madhavan deserves, in all respects, to be called the Father of Calculus and Spherical Trigonometry.Narayanan Pandit (1340–1400) Narayana Pandit , 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, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and 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, and a 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. Parameshvaran Parameshvaran was a great scientist who contributed much to Astronomy and Mathematics. He was from Vatasseri Mana on the north bank of river Nila (Bhaarathappuzha) near its mouth in a village called Aalathiyur (Aswathha Graamam). This is near the present Tirur of Malappuram district.

He was a Rigvedi (Aaswalaayanan) of Bhrigu Gothram. “Drigganitham” was his greatest contribution.The seventh century “Parahitha Ganitham” for calculations and projections in Astronomy continued its popularity for a few centuries, with some later modifications made by Mujjaalakan, Sreepathy and others, for correcting the differences found with actual occurences. But it was Parameswaran who, as a result of over fifty years of systematic observations and research on movements of celestial bodies, estimated the error factor and established a new method called Drig Sidhham as explained in his popular Drigganitham (ME 606, 1430-31 AD).

He suggested the use of “Parahitham” for “Paralokahitham” such as Thithhi, Nakshthram, Muhoortham, etc. , and his own “Drigganitham” for “Ihalokahitham” like “Jaathakam”, “Graha Moudhhyam”, “Grahanam”, etc. Unfortunately, Drigganitham Granthham has not been traced so far. Yet another of his contribution was a correction to the angle of precision of equinox mentioned by his disciple, Kelalloor Somayaaji (vide 15, below) in his “Jyothirmeemaamsa” (ch.

17). The 13 ? ° suggested by Mujjaalakan was rectified by him to 15°.There are numerous works to his credit, apart from Drigganitham. The 3-volume, 302 verse “Gola Deepika” (1443 AD) explaining about the stars and earth in very simple terms, “Jaathaka Padhhathy” in 41 verses, “Soorya Sidhhantha Vivaranam”, “Grahana Mandanam”, “Grahanaashtakam”, “Vyatheepaathaashtaka Vrththi” in 500 verses or Slokams. (The last three are believed by experts to be his works), “Aachaarya Samgraham”, “Grahana Nyaaya Deepika”, “Chandra-Chhaayaa-Ganitham”, “Vaakya Karmam” and “Vaakya Deepika” are his well-known works.He has written superb commentaries such as “Sidhhantha Deepika” on Govindaswamy’s Mahaa Bhaaskareeyam; “Karma Deepika” or “Bhata Deepika” on Aarya Bhateeyam; “Muhoortha Rathna Vyaakhyaa” on Govindaswamy’s Muhoortha Rathnam; Leelavathee Vyaakhyaa on the famous mathematical treatise, Leelavathy of Bhaaskaraachaarya-II; “Laghu Bhaaskareeya Vyaakhyaa” on Laghu Bhaaskareeyam of Bhaaskaraachaarya-I; “Jaathaka Karma Padhhathee Vyaakhyaa” on Sreepathy’s 8-chapter work on Jyothisham; the one on “Laghu Maanasam” of Mujjaalakan; “Jaathakaadesa Vyaakhyaa”; and “Prasna-Nashta Panchaasikaavrthy” also called “Paarameswari” based on the work of Prathhuyasass, son of Varaahamihiran. “Sidhhantha Deepika” contains some of his eclipse observations, including one made at Navakshethra in 1422 and two made at Gokarna in 1425 and 1430.

It also presents a mean value type formula for inverse interpolation of the sine function, a one-point iterative technique for calculating the sine of a given angle, and a more efficient approximation that works using a two-point iterative algorithm, which is essentially the same as the modern secant method. His Lilavathi Bhasya, a commentary on Bhaskara II’s Lilavathi, contains one of his most important discoveries: an early version of the mean value theorem. This is considered one of the most important results in differential calculus and one of the most important theorems in mathematical analysis, and was later essential in proving the fundamental theorem of calculus.Parameshvaran was also the first mathematician to give the radius of a circle with an inscribed cyclic quadrilateral, an expression that is normally attributed to L’Huilier (1782). Undoubtedly, there had not been many scholars of his calibre in the annals of history in the realm of Astronomy.

Nilakanthan Somayaji(1444–1544) He is one of the foremost astronomers of Kerala and considered an equal to Vatasseri Parameswaran Namboodiri, and known popularly as Kelallur Chomaathiri. He was born to Jathavedan and Arya in Kelallur (or Kerala Nallur, Kerala-Sad-Graamam in Sanskrit) Mana of Thrikkandiyur (Sree Kundapuram in Sanskrit), near Tirur, and belonged to Gaargya Gothram, Aaswalaayana Soothram of Rigvedam. Kelallur Mana later became extinct and their properties merged with Edamana Mana.They were staunch devotees at Thriprangot Siva temple. He is said to be a disciple of one Ravi who taught him Vedaantham and the basics of Astronomy and of Vatasseri Damodaran Namboodiri (son of the famous Parameswaran Namboodiri) who trained him in Astronomy and Mathematics. His most important work is “Thanthra Samgraham” (a treatise on Mathematics and Astronomy) in eight chapters with 432 verses, and apparently written in an unbelievable six days from Meenam 26 of 676 ME to Metam 1 the same year! The lucid manner in which difficult concepts are presented, the wealth of quotations, and the results of his personal investigations and comparative studies make this work a real masterpiece.

Two commentaries on this work, “Yukthi Bhaasha” (in Malayalam) by Paarangot Jyeshthhadevan Namboodiri (No. 16 below) and “Yukthi Deepika” by Sankara Varier, themselves indicate the importance of the original work. Another of his important works is a “Bhaashyam” (commentary) on “Aaryabhateeyam”. In his book “Jyorthir Meemaamsa”, he demonstrates his intellectual and scientific thinking. Some of his other works are “Chandra Chhaayaa Ganitham” (calculations relating to moon’s shadow), “Sidhhantha Darpanam” (mirror on the laws of Astronomy) and its Vyaakhyaa, “Golasaaram” (quintessence of spherical Astronomy), “Grahana Nirnayam”, “Grahanaashtakam”, “Graha Pareekshaa Kramam”, and “Sundara Raaja Prasnotharam”.

He postulated that the ratio of circumference to diameter of a circle could never be a rational number. His commentary on Aaryabhateeyam shows that his scholastic abilities extend beyond Jyothisham and Vedaantham, to the realms of Meemaamsa, Vyaakaranam and Nyaayam. Of great significance in Nilakantha’s work includes the presence of inductive mathematical proofs, a 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 ? that converges more rapidly, and the relationship between the power series of ? and arctangent.

32 He also gave sophisticated explanations of the irrationality of ? the correct formulation for the equation of the center of the planets, and a heliocentric model of the solar system. Chitrabhanu 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. 33 Born in Chovvara (Sukapuram) Graamam, Chithrabhanu Namboodiri was a mathematician and has written a Granthham titled “Eka Vimsathi Prasnothari”. It is said that Sankara Varier, another scholar (mentioned earlier) who wrote the commentary “Kriyaakramakari” was Chithrabhanu Namboodiri’s disciple.

Varier has, at several occasions, quoted his master.He 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: The achievements of such and other Kerala mathematicians were, at first, brought to the notice of scholars, both Indian and western, by Charles M Whilsh who presented a paper on the subject before the Royal Asiatic Society of Great Britain and Ireland, 3 (1835) (509 – 523). Paarangottu Jyeshthhadevan Namboodiri (1500 – 1610) jyesthadeva (c. 1500–1600) was another member of the Kerala School. He was born in Paaragottu Mana situated near Thrikkandiyur and Aalathur on the banks of river Nila.

Vatasseri Damodaran Namboodiri was his teacher.He wrote a Malayalam commentary, “Yukthi Bhaasha” for “Thanthra Sangraham” of Kelallur Neelakandha Somayaaji. It forms an elaborate and systematic exposition of calculation methods in Mathematics in its first part and Astronomy in the second part.

The treatment is in a rational and logical manner, and may turn out to be an asset to our scientific community, if properly translated and studied. He is also the author of “Drik Karanam”, a comprehensive treatise in Malayalam on Astronomy, composed in 1603 AD. His key work was the Yuktibhasa (written in Malayalam), the world’s first Calculus text. It contained most of the developments of earlier Kerala School mathematicians, particularly from Madhava.

Similar to he work of Nilakantha, it is unique in the history of Indian mathematics, in that it contains proofs of theorems, derivations of rules and series, a 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. It also contains a 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 method, and 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, and geometrical derivations of infinite series. Sankara VarmanThere remains a final Kerala work worthy of a brief mention, Sadratnamala an astronomical treatise written by Sankara Varman (1774–1839) 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 Kerala mathematics. A remarkable contribution was his computation of ? correct to 17 decimal places. Other eminent personalities who contributed to astronomy and related areas 1. Bhaaskaraachaaryan – I (early 6th century AD) Formost among Ganithajnans (astrologer / mathematician) in the entire Bhaaratham (India), Bhaskaran-I, hailed from Kerala, according to experts.In 522 AD he wrote “Mahaa Bhaaskareeyam”, also known as “Karma Nibandhham”. A Vyaakhyaanam (explanations and discussions) on Aaryabhateeyam as well as a condensed version – “Laghu Bhaaskareeyam” – of Aaryabhateeyam, have also come down to us.

(Bhaaskaraachaaryan-II who wrote “Leelaavathy” lived in the 11th century). 2. Haridathan (650 – 750 AD) Though the Aarybhata system had been followed in calculating the planetary positions, Namboothiri scholars recognised variations between the computed and observed values of longitudes of the planets. A new system called “Parahitham” was proposed by Haridathan through his famous works “Graha-Chakra-Nibandhhana” and “Mahaa-Maarga-Nibandhhana”.In 683 AD, this system was accepted throughout Kerala on the occasion of the 12-yearly Mahaamaagha festival at Thirunavaya, and is recorded in many later works. Haridathan introduced many improvements over Aarybhata system, like using the more elegant Katapayaadi system of notation in preference to the more complicated Aarybhataa’s notation. Haridathan introduced the unique system of enunciating graded tables of the sines of arcs of anomaly (Manda-jya) and of conjugation (Seeghra-jya) at intervals of 3° 45′ to facilitate the computation of the true positions of the planets.

One of the corrections introduced by Haridathan to make the Aarybhata’s results more accurate, is the “Sakaabda Samskaaram”. 3. Aadi Sankaran (788 – 820 AD)Sree Sankaran was born in Kalady in Central Kerala (nearly 50 km north east of Kochi) on the banks of river Periyar as the son of Kaippilly Sivaguru Namboothiri and Arya Antharjanam (Melpazhur Mana).

Scientific concepts naturally evolved from this highly logical and rational intellect. It is believed that Sree Sankaran was the first mathematician to moot the concept of Number Line. Ref: “Sankara Bhaashyam” (4-4-25) of the “Brihadaaranyaka Upanishad”.

It was Sree Sankaran who first expounded the idea of assigning a set of natural numbers to a straight line. As the number of elements in a set of natural numbers is infinite, it requires a symbol of infinity to represent them.A straight line can be considered to be infinitely long. Sankaran adopted a straight line as a symbol of infinity. A straight line can be divided to infinite number of parts and each of these parts can be assigned the value of a particular number. This is called number line.

Though his concept lacks the perfection of modern number line theory, Sree Sankaran exhibited his intellectual ingenuity in conceiving such a novel idea. Yet another example for Sree Sankaran’s unbiased and pure scientific pursuit of knowledge could be seen in the second “Slokam” of “Soundarya Lahari” a collection of 100 Slokams in praise of Goddess Durga written by Sree Sankaran.In the Slokam “Thaneeyaamsam paamsum thava charana pankeruhabhavam”, we can see a hint to the theory of inter-convertibility of mass and energy.

Famous scientist Albert Einstein put forward this theory much later. Einstein said mass can be converted to energy and vice-versa according to the equation E = MC? , where E = Energy released, M = Mass of the substance, and C = Velocity of light = 3 x 10?? cm/sec. In another context, Sree Sankaran postulated that the diameter of Sun is 1 lakh “Yojanas”. Later the modern scientific community calculated the diameter which agreed very closely with (just 3% error) the value provided by Sankaran. 4. Sankaranarayanan (9th century)This scholar from “Kollapuri” (Kollam) in Kerala has written a commentary (Vyaakhhyaanam) of the “Laghu Bhaaskareeyam” of Bhaaskaraachaaryan-I, titled “Sankaranaaraayaneeyam”.

The Granthham is dated 869 AD (ME 44). 5. Sreepathy (around 1039 AD) Sreepathy (Kaasyapa Gothram) has described methods for calculating the “Shadbalam” of the planets and stars. Prescribing of consequences should be based on these “Balams”.

His works include “Aarybhateeya Vyaakhhyaanams” such as “Ganitha Thilakam”, “Jaathaka Karma Padhhathi” and “Jyothisha Rathna Maala”. 6. Thalakkulathu Bhattathiri (1237 – 1295 AD) This Govindan Bhattathiri is believed to have been born in ME 412 in Thalakkulam of Aalathur Graamam, about three kilometer south of Tirur. The Illam does not exist anymore. His mother was apparently from Paazhoor.He is said to have left Keralam (to Paradesam, possibly Tamil Nadu) and studied the “Ulgranthhams” in Jyothisham under a scholar by name Kaanchanoor Aazhvaar, returned and prayed for a dozen years to Vadakkunnathan at Thrissur.

Bhattathiri’s major work is the renowned Jyothisha Granthham “Dasaadhhyaayi”. It is a majestic “Vyaakhyaanam” of the first ten chapters of the famous 26-chapter “Brihajjaathakam” in the field of Jyothissaasthram, written by Varaahamihiran of Avanthi, a sixth century scholar. Bhattathiri felt that the “Aachaaryan” had not covered anything significantly more in the rest of the chapters and therefore, left them altogether. There are also other works like “Muhoortha Rathnam” to his credit.

7. Sooryadevan This Namboothiri (Somayaaji) scholar is better known as Sooryadeva Yajwaavu. Jaathakaalankaaram” is Sooryadevan’s Vyaakhyaanam for Sreepathy’s (No. 5, above) “Jaathaka Karma Padhhathi”.

His other works include a “Laghu Vyaakhhyaanam” (simple explanation) of Aaryabhateeyam, called “Bhataprakaasam”, as well as Vyaakhhyaanams for Varaahamihiran’s “Brihadyaathra” and for Mujjaalakan’s “Laghu Maanava Karanam”. 8. Damodaran Namboodiri Damodaran Namboodiri is known for his work “Muhoorthaabharanam”.

It is believed that he had an ancestor by name Yajnan whose brother’s son, Kesavan, was a great scholar, and that Damodaran was Kesavan’s younger brother. His family is said to have belonged to a village near Thriprangod, but it is clear that it was in Taliparamba Graamam.Mazhamangalam (Mahishamangalam, vide 17, below) has recognised “Muhoorthaabharanam” as a reference work similar to “Muhoortha Rathnam” and other earlier works. 9.

Narayanan Namboodiri He has authored “Muhoortha Deepikam”. He could be the same Narayanan, one of Vatasseri Parameswaran Namboodiri’s teachers (Guru), as mentioned by Kelallur Chomaathiri (Neelakandha Somayaaji, 15, below). “Muhoortha Deepikam” is also recognised as an authoritative work, by Mazhamangalam (17, below). 10. Puthumana Somayaaji (Chomaathiri) He belonged to Puthumana Illam (Sanskritised as Noothana Graamam) of Chovvaram (Sukapuram) Graamam. He is believed to have been a contemporary of Vatasseri Namboodiri, during the 15th century AD.His famous works are “Karana Padhhathi” which is a comprehensive treatise on Astronomy in ten chapters completed in the year ME 606 (1430-31 AD), the same year as Vatasseri Namboodiri’s “Drigganitham”; “Nyaaya Rathnam”, an 8-chapter Ganitha Granthham; “Jaathakaadesa Maargam”; “Smaartha-Praayaschitham”; “Venvaarohaashtakam”; “Panchabodham”; “Grahanaashtakam”; and “Grahana Ganitham”.

To his credit is also an important mathematical equation to calculate the tangent (tan) value of an angle, as: or 11. Chennas Narayanan Namboodiripad (mid 15th century) He was considered to be an authority in the fields of Vaasthusaastram (Indian Architecture), Mathematics and Tanthram. Born in 1428, Chennas Narayanan Namboodiripad authored a book titled “Thanthra Samuchayam” which is still considered as the authentic reference manual in the field of temple architecture and rituals.In this Granthham , while elaborating on various points of Indian architectural practices, he has dealt with many mathematical principles also. The following are noteworthy.

a) | A method of arriving at a circle starting with a square, and successively making it a regular octagon, a regular 16-sided, a 32-sided, 64-sided polygons, etc. In this method some geometrical steps have been suggested. | b) | Co-ordinate system of fixing points in a plane. | c) | Converting a square to a regular hexagon having approximately equal area. | d) | Finding the width of a regular octagon, given the perimeter.

| 12. Mahishamangalam Narayanan Namboodiri (1540 – 1610) He was a member of Mahishamangalam (Mazhamangalam) Mana of Peruvanam in Thrissur district.His father Sankaran Namboothiri has written several Granthhams on Astronomy in Malayalam. Renouned scholar Sankara Varier has written a commentary “Kriyaakramakari” in Malayalam for the popular Mathematical manual “Leelavathy” (of Bhaskaraachaarya) but before commencing the 200th Slokam, he expired. It was Mahishamangalam Narayanan Namboodiri who, at the age of 18, took up the challenge of completing it. He was popularly known as “Ganitha Vith” Maths wizard.

After successfully completing “Kriyaakramakari”, Narayanan Namboodiri wrote his own commentary “Karmadeepika” for “Leelavathy”. “Upa Raaga Kriyaa Kramam” was his original work in the related topic.He has authored many Granthhams on subjects other than Astronomy, including Smaartha Praayaschitha Vimarsanam, Vyavahaara Mala ethical code of conduct, Mahishamangalam Bhaanam, Uthara Raamaayana Champu, Raasa Kreedaa Kaavyam, Raaja Ratnaavaleeyam in praise of Kerala Varma, Prince of Kochi), Daarikavadham, and Paarvatheesthuthi. 13. Mathur Nambudiripad The Granthham, “Muhoortha Padavi” (the second) is credited to Mathur Nambudiripad, whose name is not known. He has condensed the old “Muhoortha Padavi” into an amazingly short version with just 35 Slokams (verses). Since Mazhamangalam of mid-sixteenth century AD, in his “Baala Sankaram” has referred to Muhoortha Padavi, it is possible that Mathur Nambudiripad lived during the second half of the 15th century AD.

Apart from Mazhamangalam’s commentary on this Granthham, there are: a short one in Sanskrit, “Muhoortha Saranee Deepam” (author unknown); a detailed one in Sanskrit, “Varadeepika” by Purayannur Parameswaran Nambudiripad; and yet another one in Malayalam, “Muhoortha Bhaasha” by Aazhvaancheri Thampraakkal. From the above description, it is evident that the Keralite have contributed immensely towards astronomy and mathematics. It is doubtful that these contributions are rightly highlighted and presented to the younger generations. It is very important to be proud of our heritage and follow the tradition of our great heroes in these fields.

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