*VEDIC SCIENCE & ITS PRESENCE IN MODERN WORLD*

__Guigualt__*: "The Rig Veda is the most sublime conception of the great highways of humanity."*

__Max Muller__*: "In the history of the world, the Veda fills a gap which no literary work in any other language can fill."*

__Henry David Thoreau__*: What extracts from the*

**Vedas**I have read fall on me like the light of a higher and purer luminary, which describes a loftier course through purer stratum. It rises on me like the full moon after the stars have come out, wading through some far stratum in the sky."**: "Vedanta is the most impressive metaphysics the human mind has conceived."**__Alfred North Whitehead__

__J. Robert Oppenheimer__

*:*

*Access to the Vedas is the greatest privilege this century may claim over all previous centuries.*

*the amazing inventions, extraordinary ideas and world's first and advanced civilization by Indians that are still usefull today, such as plastic sergery, number system, wax, yoga, water clock, etc.*

1. India invented the Number system. Pingalacharya invented ‘zero.’ in 200 BC.

2. Indians discovered the size, shape, rotation and gravity of earth about 1000 years before Kelvin,Galileo,Newton and Copper Nicus. Aryabhatta I was the first to explain spherical shape,size ,diameter,rotaion and correct speed of Earth in 499 AD.

3. Newton’s law of Gravitational force is an ancient Indian discovery. In Siddhanta Siromani ( Bhuvanakosam 6 ) Bhaskaracharya II described about gravity of earth about 400 years before Sir Isaac Newton.

4. Bhaskaracharya II discovered Differential calculus.

5. Theory of Continued Fraction was discovered by Bhaskaracharya II.1. India invented the Number system. Pingalacharya invented ‘zero.’ in 200 BC.

2. Indians discovered the size, shape, rotation and gravity of earth about 1000 years before Kelvin,Galileo,Newton and Copper Nicus. Aryabhatta I was the first to explain spherical shape,size ,diameter,rotaion and correct speed of Earth in 499 AD.

3. Newton’s law of Gravitational force is an ancient Indian discovery. In Siddhanta Siromani ( Bhuvanakosam 6 ) Bhaskaracharya II described about gravity of earth about 400 years before Sir Isaac Newton.

4. Bhaskaracharya II discovered Differential calculus.

5. Theory of Continued Fraction was discovered by Bhaskaracharya II.

6. The place value system, the decimal system was developed in India in 100 BC.

7. Indians discovered Arithmetic and Geometric progression. Arithmetic progression is explained in Yajurveda.

8. Govindaswamin discovered Newton Gauss Interpolation formula about 1800 years before Newton.

9. Vateswaracharya discovered Newton Gauss Backward Interpolation formula about 1000 years before Newton.

10. Madhavacharya discovered Taylor series of Sine and Cosine function about 250 years before Taylor.6. The place value system, the decimal system was developed in India in 100 BC.

7. Indians discovered Arithmetic and Geometric progression. Arithmetic progression is explained in Yajurveda.

8. Govindaswamin discovered Newton Gauss Interpolation formula about 1800 years before Newton.

9. Vateswaracharya discovered Newton Gauss Backward Interpolation formula about 1000 years before Newton.

10. Madhavacharya discovered Taylor series of Sine and Cosine function about 250 years before Taylor.

11. Madhavacharya discovered Newton Power series.

12. Madhavacharya discovered Gregory Leibnitz series for the Inverse Tangent about 280 years before Gregory.

13. Madhavacharya discovered Leibnitz power series for pi about 300 years before Leibnitz.

14. Parameswaracharya discovered Lhuiler’s formula about 400 years before Lhuiler.

15. Nilakanta discovered Newton’s Infinite Geometric Progression convergent series.

16. Theorems relating the diameter,volume and circumference of circles discovered by Madhavacharya, Puthumana Somayaji, Aryabhatta, Bhaskaracharya…….

17. The value of pi was first calculated by Aryabhatta I in 499 AD,ie more than 1350 years before Lindemann

18. Boudhayana discovered Pythagorus Theorem in 800BC. ie 300 years before Pythagorus.

19. Algebra, trigonometry and calculus came from India. Quadratic equations were by Sridharacharya in the 11th Century.

20. While the Greeks were using only upto a maximum value 1000, Indians could go upto 18th power of 10 level during Vedic period.

21. Infinity was well known for ancient Indians. BhaskaracharyaII in Beejaganitha

(stanza-20) has given clear explanation with examples for infinity

22. Positive and Negative numbers and their calculations were explained first by Brahmagupta in his book Brahmasputa Siddhanta.

23. Sterling formula was discovered by Brahmagupta about 1000 years before Sterling.

24. Demovier’s theorem of positive integral was discovered by Brahmagupta in 628 A.D, i.e around 1000 years before Demovier.

25. Puthumana Somayaji discovered Demovier’s infinite series in 1140 AD,i.e more than 200 years before Demovier.

26. Maharshi Sushruta is the father of surgery. 2600 years ago he and health scientists of his time conducted surgeries like cesareans, cataract, fractures and urinary stones. Usage of anesthesia was well known in ancient India. He was the first person to perform plastic surgery.

27. When many cultures in the world were only nomadic forest dwellers over 5000 years ago, Indians established Harappan culture in Sindhu Valley (Indus Valley Civilization).

28. The world’s first University was established in Takshila in 700BC. More than 10,500 students from all over the world studied more than 60 subjects. The University of Nalanda built in the 4th century BC was one of the greatest achievements of ancient India in the field of education.

29. According to the Forbes magazine, Sanskrit is the most suitable language for computer software.

30. Ayurveda is the earliest school of medicine known to humans.

31. Although western media portray modern images of India as poverty stricken and underdeveloped through political corruption, India was once the richest empire on earth.

32. According to the Gemmological Institute of America, until 1896, India was the only source of diamonds to the world.

33. USA based IEEE has proved what has been a century-old suspicion amongst academics that the pioneer of wireless communication was Professor Jagdeesh Bose and not Marconi.

34. The earliest reservoir and dam for irrigation was built in Saurashtra.

35. Chess was invented in India.

36. The first philosopher who formulated ideas about the atom in a systematic manner was Kanada who lived in the 6th century B.C.

37. All the atomic reactors in the world are in Shiva Linga Shape which is an Indian contribution.

38. Padanjali maharshi discovered Sound waves.

39. Yoga is an ancient Indian gift to the world.

40. Shayanacharya discovered velocity of light.

41. Maharshi Bharadwaja discovered different types of light rays.

42. Maharshi Bharadwaja was the first person to give definition about aeroplane. He explained about different types aeroplanes in his book “Vimana Thantra” about 2000 years before Right Brothers.

43. Maharshi Bharadwaja discovered spectrometer. In his “Yantra Sarvaswa” he explained about more than 100 instruments.

44. The different colours of light, VIBGYOR are mentioned in Rigveda which was written more than 6000 years ago.

45. Maharshi Charaka discovered Psychology and Quantum healing system.

46. Varahamihira discovered the concept of “Budding of plants”.

47. Varahamihira discovered Comets in 505 AD, i.e more than 1100 years before Haley.

48. Gouthama Maharshi discovered the wave nature of sound about 1400 years before Hyghen.

49. Seven continents are mentioned in Padmapurana.

*Kak [1] notes that Sayana, Prime-minister of India in the 14th century, could decipher an extract from The Rigveda from which followed that the value of the velocity of light was equal to 300,000 km/s. Note that in Europe, Danish astronomer O. RÃ¶mer could measure the same value of the velocity of light only in 1676, i.e. around 150 year later.

*The contributions of the Namboothiris in Astrology, Astronomy and Mathematics have been immense. They had a capacity for unmistakable and sharp observations on the natural phenomena and accurate ability of deducting complicated theoretical formulae. The works of about 20 prominent ones among them during a long period of about a millenium between the seventh and the eighteenth century (AD) are enumerated here.

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).

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 (Click here) 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".

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 = MC², where E = Energy released, M = Mass of the substance, and C = Velocity of light = 3 x 10¹Âº 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.

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).

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".

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.

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".

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 treacher 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).

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 p 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.

sine (A + B) = sine A cos B + cos A sine B

This is known as "Jeeve Paraspara Nyaaya".

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. For a detailed appreciation of his contribution, refer to the excellent paper of R G Gupta,"Second Order of Interpolation of Indian Mathematics", Ind, J.of Hist. of Sc. 4 (1969) 92-94.

Again Madhavan provides the power series expansions for sin x and cos x for an arc x correct to 1/3600 of a degree.

Vatasseri 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 ½° 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.

Undoubtedly, there had not been many scholars of his calibre in the annals of history in the realm of Astronomy.

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.

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).

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

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.

He is one of the teachers of Kelallur Chomaathiri, and was a scholar in both Astronomy and Vedaantham. His treatise "Aachaara Deepika" is on Jyothisham.

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. According to Ulloor, he lived during 1465 and 1545 (roughly), though according to another version, he was born on June 17, 1444 on a Wednesday.

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.

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.

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.

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.

One Narayanan has written a commentary on Bhaaskaraachaaryan's Leelaavathy, which has been variously referred to as "Karmadeepika", "Karmadeepakam" and "Kriyaakramakari". The work is well-focussed and neither too elaborate nor too short.

Another of his works is " Karmasaaram" which discusses "Grahasphhutaanayanam" and other aspects of the Drik tradition. It is in four chapters and may have been written during the second half of the 16th century AD.

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.

Chithrabhanu Nambudiri's "Eka Vimsathi Prasnothari" gives a method of solving the binomials (A + B), (A - B), (A² + B²), (A³ + B³), (A³ - B³), AB, etc. Given any two of these, the book gives twentyone different ways to solve for A and B. As he is believed to be the master of Sankara Varier, his period could be 16th century.

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).

BIBLIOGRAPHY

Sanskrit:

1. Aryabhatiya of Aryabhata with Nilakanta Somasutvan's Com. Ed. Pub. in 3 parts by K Sambasiva Sastri. Trivandrum, 1977.

2. Drigganitham of Parameswara. Cr. Ed. By K V Sarma, Vishveshvaranand Vedic Research Institute, Hoshiarpur, 1963.

3. Goladipika of Parameswara. Ed. Tr. K V Sarma, Madras, 1956 - 57.

4. Grahananyayadipika. Cr. Ed. Tr. K V Sarma, V V R I , Hoshiarpur, 1966.

5. Grahanashtaka of Parameswara. Ed. Tr. K V Sarma, Madras, 26 Parts(I-IV),47-60,1961.

6. Jyothirmimamsa of Nilakantha Somayaji. Ed. K V Sarma, V V B I S, Hoshiarpur, 1977.

7.Tantrasangraha of Nilakantha Somayaji. Cr. Ed. K V Sarma, V V B I S & I S, Hoshiarpur, 1977.

8. Sphutachandrapti of Madhava. K V Sarma, V V I, Hoshiarpur, 1973.

English:

1. Rajagopal C T and Venkatarama A - The sine and cosine series. J. Asiatic Soc. of Bengal. 3rd Series. 15 pp. 1 - 13, 1949.

2. Rajagopal C T and Aiyar T V Vedamurthy - On the Hindu Proof of Gregory Series. Scripta Mathematica 17, Nos.1-2, pp 65-74, 1951.

3. Sarma K V-A History of the Kerala School of Hindu Astronomy. VVI, Hoshiarpur, 1972.

4. Swarup G, Bag A K and Shukla K S - History of Oriental Astronomy. University Press, Cambridge, 1987.

5. Krishnan Namboodiri, Chekrakkal (Dr) - PhD Thesis

Malayalam:

1. Rao Sahib, Mahakavi Ulloor S Parameswara Aiyer - Kerala Sahitya Charitram, Vol. 1.(4th Ed.) 1974 & Vol. 2 (4th Ed.) 1979; Published by Department of Publications, University of Kerala, Thiruvananthapuram.

11. Madhavacharya discovered Newton Power series.

12. Madhavacharya discovered Gregory Leibnitz series for the Inverse Tangent about 280 years before Gregory.

13. Madhavacharya discovered Leibnitz power series for pi about 300 years before Leibnitz.

14. Parameswaracharya discovered Lhuiler’s formula about 400 years before Lhuiler.

15. Nilakanta discovered Newton’s Infinite Geometric Progression convergent series.

16. Theorems relating the diameter,volume and circumference of circles discovered by Madhavacharya, Puthumana Somayaji, Aryabhatta, Bhaskaracharya…….

17. The value of pi was first calculated by Aryabhatta I in 499 AD,ie more than 1350 years before Lindemann

18. Boudhayana discovered Pythagorus Theorem in 800BC. ie 300 years before Pythagorus.

19. Algebra, trigonometry and calculus came from India. Quadratic equations were by Sridharacharya in the 11th Century.

20. While the Greeks were using only upto a maximum value 1000, Indians could go upto 18th power of 10 level during Vedic period.

21. Infinity was well known for ancient Indians. BhaskaracharyaII in Beejaganitha

(stanza-20) has given clear explanation with examples for infinity

22. Positive and Negative numbers and their calculations were explained first by Brahmagupta in his book Brahmasputa Siddhanta.

23. Sterling formula was discovered by Brahmagupta about 1000 years before Sterling.

24. Demovier’s theorem of positive integral was discovered by Brahmagupta in 628 A.D, i.e around 1000 years before Demovier.

25. Puthumana Somayaji discovered Demovier’s infinite series in 1140 AD,i.e more than 200 years before Demovier.

26. Maharshi Sushruta is the father of surgery. 2600 years ago he and health scientists of his time conducted surgeries like cesareans, cataract, fractures and urinary stones. Usage of anesthesia was well known in ancient India. He was the first person to perform plastic surgery.

27. When many cultures in the world were only nomadic forest dwellers over 5000 years ago, Indians established Harappan culture in Sindhu Valley (Indus Valley Civilization).

28. The world’s first University was established in Takshila in 700BC. More than 10,500 students from all over the world studied more than 60 subjects. The University of Nalanda built in the 4th century BC was one of the greatest achievements of ancient India in the field of education.

29. According to the Forbes magazine, Sanskrit is the most suitable language for computer software.

30. Ayurveda is the earliest school of medicine known to humans.

31. Although western media portray modern images of India as poverty stricken and underdeveloped through political corruption, India was once the richest empire on earth.

32. According to the Gemmological Institute of America, until 1896, India was the only source of diamonds to the world.

33. USA based IEEE has proved what has been a century-old suspicion amongst academics that the pioneer of wireless communication was Professor Jagdeesh Bose and not Marconi.

34. The earliest reservoir and dam for irrigation was built in Saurashtra.

35. Chess was invented in India.

36. The first philosopher who formulated ideas about the atom in a systematic manner was Kanada who lived in the 6th century B.C.

37. All the atomic reactors in the world are in Shiva Linga Shape which is an Indian contribution.

38. Padanjali maharshi discovered Sound waves.

39. Yoga is an ancient Indian gift to the world.

40. Shayanacharya discovered velocity of light.

41. Maharshi Bharadwaja discovered different types of light rays.

42. Maharshi Bharadwaja was the first person to give definition about aeroplane. He explained about different types aeroplanes in his book “Vimana Thantra” about 2000 years before Right Brothers.

43. Maharshi Bharadwaja discovered spectrometer. In his “Yantra Sarvaswa” he explained about more than 100 instruments.

44. The different colours of light, VIBGYOR are mentioned in Rigveda which was written more than 6000 years ago.

45. Maharshi Charaka discovered Psychology and Quantum healing system.

46. Varahamihira discovered the concept of “Budding of plants”.

47. Varahamihira discovered Comets in 505 AD, i.e more than 1100 years before Haley.

48. Gouthama Maharshi discovered the wave nature of sound about 1400 years before Hyghen.

49. Seven continents are mentioned in Padmapurana.

*Kak [1] notes that Sayana, Prime-minister of India in the 14th century, could decipher an extract from The Rigveda from which followed that the value of the velocity of light was equal to 300,000 km/s. Note that in Europe, Danish astronomer O. RÃ¶mer could measure the same value of the velocity of light only in 1676, i.e. around 150 year later.

*The contributions of the Namboothiris in Astrology, Astronomy and Mathematics have been immense. They had a capacity for unmistakable and sharp observations on the natural phenomena and accurate ability of deducting complicated theoretical formulae. The works of about 20 prominent ones among them during a long period of about a millenium between the seventh and the eighteenth century (AD) are enumerated here.

**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 (Click here) 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 = MC², where E = Energy released, M = Mass of the substance, and C = Velocity of light = 3 x 10¹Âº 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. Irinjaatappilly Madhavan Namboodiri (1340 - 1425)8. Irinjaatappilly Madhavan Namboodiri (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 treacher 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).

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 p 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.

sine (A + B) = sine A cos B + cos A sine B

This is known as "Jeeve Paraspara Nyaaya".

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. For a detailed appreciation of his contribution, refer to the excellent paper of R G Gupta,"Second Order of Interpolation of Indian Mathematics", Ind, J.of Hist. of Sc. 4 (1969) 92-94.

Again Madhavan provides the power series expansions for sin x and cos x for an arc x correct to 1/3600 of a degree.

**9. Vatasseri Parameswaran Namboodiri (1360 - 1455)**Vatasseri 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 ½° 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.

Undoubtedly, there had not been many scholars of his calibre in the annals of history in the realm of Astronomy.

**10. 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.

**11. 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).

12. Puthumana Somayaaji (Chomaathiri)12. 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

**13. 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.

**14. Ravi Namboodiri**He is one of the teachers of Kelallur Chomaathiri, and was a scholar in both Astronomy and Vedaantham. His treatise "Aachaara Deepika" is on Jyothisham.

**15. Kelallur Neelakandha Somayaaji (1465 - 1545)**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. According to Ulloor, he lived during 1465 and 1545 (roughly), though according to another version, he was born on June 17, 1444 on a Wednesday.

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.

**16. Paarangottu Jyeshthhadevan Namboodiri (1500 - 1610)**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.

**17. 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.

**18. 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.

**19. Narayanan Namboodiri**One Narayanan has written a commentary on Bhaaskaraachaaryan's Leelaavathy, which has been variously referred to as "Karmadeepika", "Karmadeepakam" and "Kriyaakramakari". The work is well-focussed and neither too elaborate nor too short.

Another of his works is " Karmasaaram" which discusses "Grahasphhutaanayanam" and other aspects of the Drik tradition. It is in four chapters and may have been written during the second half of the 16th century AD.

**20. Chithrabhanu Namboodiri (16th century)**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.

Chithrabhanu Nambudiri's "Eka Vimsathi Prasnothari" gives a method of solving the binomials (A + B), (A - B), (A² + B²), (A³ + B³), (A³ - B³), AB, etc. Given any two of these, the book gives twentyone different ways to solve for A and B. As he is believed to be the master of Sankara Varier, his period could be 16th century.

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).

BIBLIOGRAPHY

Sanskrit:

1. Aryabhatiya of Aryabhata with Nilakanta Somasutvan's Com. Ed. Pub. in 3 parts by K Sambasiva Sastri. Trivandrum, 1977.

2. Drigganitham of Parameswara. Cr. Ed. By K V Sarma, Vishveshvaranand Vedic Research Institute, Hoshiarpur, 1963.

3. Goladipika of Parameswara. Ed. Tr. K V Sarma, Madras, 1956 - 57.

4. Grahananyayadipika. Cr. Ed. Tr. K V Sarma, V V R I , Hoshiarpur, 1966.

5. Grahanashtaka of Parameswara. Ed. Tr. K V Sarma, Madras, 26 Parts(I-IV),47-60,1961.

6. Jyothirmimamsa of Nilakantha Somayaji. Ed. K V Sarma, V V B I S, Hoshiarpur, 1977.

7.Tantrasangraha of Nilakantha Somayaji. Cr. Ed. K V Sarma, V V B I S & I S, Hoshiarpur, 1977.

8. Sphutachandrapti of Madhava. K V Sarma, V V I, Hoshiarpur, 1973.

English:

1. Rajagopal C T and Venkatarama A - The sine and cosine series. J. Asiatic Soc. of Bengal. 3rd Series. 15 pp. 1 - 13, 1949.

2. Rajagopal C T and Aiyar T V Vedamurthy - On the Hindu Proof of Gregory Series. Scripta Mathematica 17, Nos.1-2, pp 65-74, 1951.

3. Sarma K V-A History of the Kerala School of Hindu Astronomy. VVI, Hoshiarpur, 1972.

4. Swarup G, Bag A K and Shukla K S - History of Oriental Astronomy. University Press, Cambridge, 1987.

5. Krishnan Namboodiri, Chekrakkal (Dr) - PhD Thesis

Malayalam:

1. Rao Sahib, Mahakavi Ulloor S Parameswara Aiyer - Kerala Sahitya Charitram, Vol. 1.(4th Ed.) 1974 & Vol. 2 (4th Ed.) 1979; Published by Department of Publications, University of Kerala, Thiruvananthapuram.