
It is said that a Kerala Braahmanan by name Achyuthan performed "Thapas" to Brihaspathi, who appeared and asked what favours he wanted. He asked for Brihaspathi's condensed version of the 2,000 "Jaathaka Skandhham" portion of the Jyothissaasthram written by Sree Naaraayanan and comprising of four lakh Granthhams. He then prayed and pleased both Sukran and Sree Parameswaran and obtained their 1,000 and 2,000 Granthhams respectively, and taught the Guru Matham, Sukra Matham and Saambasiva Matham to his disciples.
The 7th century (AD) witnessed tremendous development in Jyothissaasthram in Kerala. Siksha, Vyaakaranam, Niruktham, Jyothisham, Kalpam and Chhandovichithi are the six "limbs" (Shadaangams) of Vedam. Jyothisham in those days was used for determining auspicious times for various Vaidika Karmams (religious rituals).
Jyothissaasthram has three Skandhhams (branches)  Ganitham, Samhitha and Hora. In addition, it is seen to have six Angams (parts)  Jaathakam, Golam, Nimitham, Prasnam, Muhoortham and Ganitham. Of these, Golam and Ganitham are in Ganitha Skandhham, Jaathakam, Prasnam and Muhoortham in Hora Skandhham, and Nimitham in Samhitha Skandhham. Nimitham is partly covered in Hora also.
Some consider Jyothissaasthram to consist of two parts  Pramaanam and Phhalam with Ganitha Skandhham discussing the Pramaanam part and the other two Skandhhams, the Phhalam part. The former includes Soorya and Chandra Grahanams (solar and lunar eclipses), the Mouddhyam of Grahams (stars and planets), Chandra Sringonnathi (lunar cycles), and the Gathi Bhedams (changes in motion) of planets, and the methods of their prediction, and also descriptions of Bhoogola Khagolams (earth, planets and stars). Whereas, Jaathakam, Prasnam, Bhootha Sakunaadi Lakshanams (omens, etc.), Muhoorthams (auspicious days/ times), etc. are included in the Phhalam part. Of these, Jaathakam and Prasnam are extremely important. Jaathakam (horoscope) involves predicting the good and the bad events during the entire life of a person based on the position of the planets and the stars at the precise time of his / her birth. Prasnam predicts the good and bad results for the subject again based on the planetary and star positions at the time of some special events/ tests proposed to be undertaken by the subject, usually the learned and the pious.
CONTRIBUTIONS OF NAMBOOTHIRIS TO ASTRONOMY AND MATHEMATICS
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), BhaskaranI, 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.
(BhaaskaraachaaryanII 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 "GrahaChakraNibandhhana" and "MahaaMaargaNibandhhana". In 683 AD, this system was accepted throughout Kerala on the occasion of the 12yearly 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 (Mandajya) and of conjugation (Seeghrajya) 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" (4425) 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 interconvertibility of mass and energy. Famous scientist Albert Einstein put forward this theory much later. Einstein said mass can be converted to energy and viceversa according to the equation E = MC², where E = Energy released, M = Mass of the substance, and C = Velocity of light = 3 x 10¹º 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 BhaaskaraachaaryanI, 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 26chapter "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)
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 moonsentences, "Aganitham", an extensive treatise on the computation of planets, "Golavaadam", "SphhutaChandraapthi", "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 ParasparaNyaaya
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) 9294.
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, 143031 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 3volume, 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", "ChandraChhaayaaGanitham", "Vaakya Karmam" and "Vaakya Deepika" are his wellknown 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 BhaaskaraachaaryaII; "Laghu Bhaaskareeya Vyaakhyaa" on Laghu Bhaaskareeyam of BhaaskaraachaaryaI; "Jaathaka Karma Padhhathee Vyaakhyaa" on Sreepathy's 8chapter work on Jyothisham; the one on "Laghu Maanasam" of Mujjaalakan; "Jaathakaadesa Vyaakhyaa"; and "PrasnaNashta 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)
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 (143031 AD), the same year as Vatasseri Namboodiri's "Drigganitham"; "Nyaaya Rathnam", an 8chapter Ganitha Granthham; "Jaathakaadesa Maargam"; "SmaarthaPraayaschitham"; "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 in the inverse form,
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 16sided, a 32sided, 64sided polygons, etc. In this method some geometrical steps have been suggested. 
b)  Coordinate 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, KeralaSadGraamam 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 midsixteenth 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 wellfocussed 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
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Article
prepared by:
1. Dr C Krishnan Namboodiri, Chekrakkal Illam, Panchama School Road, Calicut  673001, Phone: 0495361361 & 2. Dr P G Namboodiri, "Swasthi" Mana, Tali, Calicut  673002, Phone: 0495702665 English
translation:

