Nachiketha: What is mind?
Yamadharma: The charioteer of five-horsed chariot.
Nachiketa: And who is the traveler?
Yamadharma: The soul
Nachiketha: What is soul?
Yamadharma: that you have to find yourself through self-realization
Mind is the true laboratory where behind illusions we uncover the laws of truth. It is in this context that we introduce one of the greatest Indian mathematicians who lived in Kerala during the middle ages. Astronomical and mathematical development in India is not well known after 10th century AD and many important works that prove the presence of many great astronomers and mathematicians of Kerala remain little known in the field of history of science. One such great astronomer-mathematician was Madhava of Sangama grama. Sangamagrama Madhava’s work Venuaroham was used for the computation of the true longitude of the moon. Sangamagrama Madhava was one of the great geniuses of the time before Keplar and Newton. He lived in Irinjalakkuda in the 14th century AD. Among his works, only Venuaroham and Sphutachandrapti are available in print and the rest are known through references made by his shishyas. In the following sections, we introduce some of the works of Sangamagrama Madhava and members of the Kerala School.
While Europe was divided in modeling solar system to geocentric and heliocentric systems even in the 17th century, an astronomer-mathematician from Kerala during the 16th century described observational studies as geocentric which can be transformed into a mathematical model with the sun as the centre. Neelakanda Somayaji, a 16th century astronomer-mathematician belonging to a long guru-sishya parampara chain of five hundred years of length, describes in his work, Tantra Sangraha, the subject of observational astronomy along with necessary mathematical techniques.
It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian astronomers and mathematicians over many hundreds of years divided into ancient, classical, medieval and modern periods. Chief among astronomers-mathematicians from each period are: Ancient: Apastamba, Baudhayana, Katyayana, Manava, Panini, Pingala and Yajnavalkya; Classical: Vararuchi, Aryabhata, Varahamihira, Brahmagupta; Medieval: Narayana Pandita, Bhaskaracharya, Sangamagrama Madhava, Nilakanda Somayaji, Jyestadeva, Achuta Pisharoti, Melpathur Narayan Bhattathiri, Sankaravarman; and Modern: Srinivas Ramanujan, Harish Chandra, Narendra Karmakar, S Chandrasekhar, SN Bose.
The beautiful number system (zero and decimal system) invented by the Indians on which mathematical development has rested is complimented by Laplace as: ‘The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of antiquity, Archimedes and Apollonius. It was Einstein who said we should be grateful to Indians who taught us how to count.’
While the rest of the world was in the dark ages, India made strides in Mathematics. The last 3000 years of legacy runs to this day, through the works of Sulbakaras (800-600 BC), Aryabhata, Varahamihira, Brahmagupta, Bhaskaracharya, Sangamagrama Madhava, Nilakanda Somayaji, Jyeshtadeva, Sankaravarman extending to those of Srinivasa Ramanujan, SN Bose, Harish Chandra, Prasanta Chandra Mahalanobis, and reaching to the current period of Narendra Karmakar, Jayan Narlikar, SR Srinivasa Varadhan, ECG Sudarsan and Thanu Padmanabhan.
Political chaos caused halting of further generation of new knowledge in North India while Kerala, at the south-western tip of India, escaped the majority of such political upheaval, allowing a generally peaceful atmosphere for the ‘uninterrupted’ pursuit of scientific development. It is hailed as the second Golden Age of Indian Mathematics, first being the period of 5th century AD to 10th Century AD. It has come to light only during the last few decades of the 20th century that mathematics (and astronomy) continued to flourish in Kerala for several hundred years during the medieval era, especially from the 14th-18th century. Kerala mathematics was strongly influenced by astronomy leading to the derivation of mathematical results of very high importance. As a result of the untiring works of people like Prof KV Sharma, who found that only about 1% of the total available manuscripts in mathematics and astronomy in Kerala is deciphered and made known to the world while the rest is still under the vast unexplored ocean of knowledge. It is quite probable that there are still further discoveries of ‘Kerala mathematics’ to be made, and a full analysis has yet to be carried out even though several findings have already been showed that several major concepts of renaissance European mathematics attributed to stalwarts like Newton, Leibnitz, Gregory, Taylor and Euler were first developed in India. This further demands the necessity of mining out the unexplored landscapes of Kerala mathematics so that we may be lucky enough to get gems of high values and qualities. In this context we should remember a self-taught mathematician from Trivandrum, P Padmakumar, who discovered astonishing properties of the magic square called Srirama Chakram in respect of number theory and astronomy, which is now known as Magic Square. We should promote the works of such people among us who are capable of carrying out such wonderful jobs of deciphering our ancient knowledge.
Sangamagrama Madhava
Of all the mathematicians of medieval period, name of Sangamagrama Madhava is the most important who founded a continuous chain of Guru Shishya parampara from the 14th century to the 18th century, and is generally known as Kerala School of Mathematics. Sangamagrama Madhava and his school were known to the western world through a series of papers published by Charles Whish in 1834 in the journal called Transactions of Asiatic Society of Great Britain and Ireland. In his series of papers, Whish showed that works of Newton, Leibnitz, Gregory and others (who lived during 17th -18th century) were just rediscoveries of the mathematics contributed by Kerala School. However, his works did not get much attention from the academicians and researchers of the west. Only after one century of Whish’s works that the world started knowing and admiring the valuable contributions of Kerala Mathematics thorough Prof SK Sharma, C Rajagopal and his colleagues.
Sangamagrama Madhava’s Origins
From a quartet from one of the works of Madhava (which is the only available work of Madhava), namely Venuaroham, his place of birth is in Kallettinkara village in Irinjalakkuda, which is about 2 km from Irinjalakkuda Railway station. From the following sloka (quartet) from Venuaroham, it is clear that the mathematician-astronomer was born to the family known as Irinjarappally and his given name was Madhava.
Bakuladhishtithathwena / Viharo yo visishyathe
Gruhanamani soyam syath / Nijanamani Madhava (Venuaroham, 13th sloka)
(I, bearing the name Madhav, was born to the family bearing the house name (vihara with a bakula tree nearby) Irinjarappally. (There is a family residing in Kallettinkara with the name Irinjarappally.)
The family temple of Madhava with Sree Krishna as the deity is near the Irinjarappally house. We can see two granite blocks in the campus of the temple which were used by Madhava , one for meditation and the other to observe the night sky by lying down on the granite bed. (Please see figures)
It is a matter of concern that the present generation in Irinjalakkuda is not aware of this great mathematician-astronomer who worked under the same sky and walked the same piece of earth. With the help of Cochin University of Science and Technology, we have conducted a series of workshops and seminars in Irinjalakkuda where experts in the field of Mathematics, Physics and Astronomy took classes which benefited nearby school and college students. With the help of a local college, a documentary on Sangamagrama Madhava was produced for the benefit of the public.
One of the members of the Kerala School, namely Jyeshtadeva needs a special mention. While the rest of the scholars wrote their works in Sanskrit, Jyeshta Deva wrote his book, Yukthi Bhasha, a treatise in mathematics and astronomy, in Malayalam for wider dissemination of the knowledge. Attempts are being made by experts to decipher the book Yukthi Bhasha in the light of modern mathematics. The mathematical and astronomical studies carried out by Kerala School must be studied and explored.
Some of the Main Contributions of Sangamagrama Madhava
We know that one of the major contributions of Indian Mathematics is the concept of Zero and the decimal number system. One cannot pinpoint the discovery of Zero to any particular person. Another important contribution to the world of mathematics is the concept of infinity imported to mathematics, the credit of which goes to Sangamagrama Madhava. He was able to show that one can get a finite value by adding infinite terms or a finite value can be expressed as an infinite series. Usually, this discovery is attributed to Gregory, Newton, Taylor and others. Historians of mathematics have started acknowledging this fact and have started renaming some of the well-known series as Madhava-Gregory Series, Madhava-Newton Series, Madhava-Taylor Series, etc.
It is quite interesting to note that both the concepts of zero and infinity are contributions of India which influenced the Indian systems of Philosophy to a great extent. The concept of infinity could have been there in the mind of Indian Philosophers. That is why, we have a sloka in Isavasyopanisha which means:
That is infinite, this is infinite, when infinity is added to infinity, infinity remains and when infinity is taken from infinity, infinity remains.
This is true for zero also. No wonder that Indians represent the infinite extension of the sky with number zero in Bootasankhya representation of numbers.
Madhava tamed the infinity to generate finite values by adding infinite terms. Some of the infinite series discovered by Madhava were rediscovered by European mathematicians about two to three centuries later as detailed by the Cambridge University Professor of Mathematics George Gheverghese Joseph. Some of such infinite series discovered by Madhava are Madhava- Gregory series, Madhava-Newton series, Madhava-Leibnitz series, Madhava -Gregory-Leibnitz series, and Madhava-Taylor series.
The discovery of infinite series in trigonometry for sine and cosine alone is sufficient to consider Sangamagrama Madhava as equal to Newton, Leibnitz, Lagrange, Laplace, etc. Another important contribution by Madhava is the representation of the irrational number ‘pi’ as an infinite series.
Madhava used the infinite series formula to evaluate the value of pi correctly to 11 decimal places as 3.14159265359. Recent studies show that calculus, an important branch of Modern Mathematics, originated in Kerala School well before the time of Newton and Leibnitz. In Jyestadeva’s Yukthi Bhasha, which dates hundred years before the time of Newton and Leibnitz, we find the formulae for integration and differentiation. It is said that Yukthi Bhasha is the first text book in the world dealing with Calculus .
Table of Trigonometric Term Sine of Angle
Another wonderful contribution of Sangamagrama Madhava is his table for Sine of an angle from zero to 90 degrees at an interval of 3.75 degrees. The values are represented in word numerals called katapayadi number system invented by Vararuchi during the 5th century AD. The table is represented as sloka as given below:
For example sin (7.5) has the value हिमाद्रिर्वेदभावनः (himādrirvēdabhāvanaḥ =0.13052623 and modern value for sin 7.5 is 0.13052619!
Venuaroham
Venuaroham is one of the most important works of Madhava which describes the true positions (longitudes) of the moon in the sky. Information about the moon’s correct position is needed since time for yajnas, poojas, etc. are calculated based on this knowledge. Calculations are based on the anomalistic cycle of the moon around the earth.
Anomalistic Cycle used by Sangamagrama Madhava
The fundamental lunar cycles in relation to the Earth are the Synodic cycle, which has a period of 29.5 days (New Moon to New Moon) and the Anomalistic cycle (perigee to perigee) which is 27 days 13hrs 18min 34.45s (about 27.5 days). Anomalistic cycles from a zero epoch will end respectively at cycle no., days, h, m, s as follows (for example the first cycle 1 ends at 27d , 13 h, 18min, 34.45 sec) :
1, 27,13 18 34.45
2, 55, 02,37, 08.90
3, 82, 15, 55, 43.35
4, 110, 05, 14, 17.79
5, 137, 18, 32, 52.24
6, 165, 07, 51, 36.69
7, 192, 21, 10, 01.14
8, 220, 10, 38, 35.59
9, 247, 23, 47, 10.04
Note that the difference between successive cycles is about 12 hours and alternate cycles is about 24 hours . This means that the successive durations are alternately corresponding to day-night, day-night difference.
Nine cycles constitute nearly 248 days and the difference in longitudes of successive days (delta lambda) constitute the chandravakyas developed by Madhava based on the katapayadi number system.
The series of vakyas begins from the moment when the moon is at apogee (Chandra thunga yogam) and each vakya corresponds to successive day’s longitude of the moon.
Algorithm of Madhava
Position of the moon and the apogee coincide during a time called Dhruvam or dhruva kalam. The algorithm of Madhava is based on Computation of 9 dhruvas (D1, … D9) using Madhava’s constant and Kalidina K. When position of the moon and the apogee coincide during a time, Dhruvam or dhruva kalam, from that point of time there will be 9 Chandra thunga yogam which can be computed as follows. For each moon revolution around the earth, the Chandra thunga yogam will get shifted by 3 degrees every day. To start with a reference point, the starting point will be when Chandra thunga yogam happens at sunrise (Suryodaya Madhya). How many days have completed when the Chandra thunga yogam takes place at Suryodaya Madhya. The number of days of this type is 188611 and the completed moon’s orbiting around the earth is 6845. From this we can calculate the moon orbiting period as 188611/ 6845 = 27.5 days approximately (27 days 13h 18m 34.45s to be exact). Based on Madhava’s algorithm, we can find that anomalistic cycles from a zero epoch will end respectively at Dhruvas B1 to B9 at 6460, 3411, 362, 4158, 1109, 4905, 1856, 5652, and 2603 respectively and corresponding chandravakya S1 to S9 as 0, 28, 56, 83, 111, 138, 166, 193, and 221respectively.
We can represent chandravakyas with corresponding Dhruvas as shown in the figure. Vakyas are arranged in ascending order while corresponding dhruvas are in descending order. The successive difference between chandravakyas is 28 appearing on the day side on the right and night side on the left. This is like branches of a bamboo stem (Venu) going up alternately left and right in a bamboo pole or venu pole and hence the method of Sangamagrama Madhava to evaluate position of Moon in the sky is known as VENUAROHAM.
Conclusion
In the above sections, we presented a bird’s eye view of the wonderful works carried out by Sangamagrama Madhava and his school. In order to expand the present work being carried out, it is proposed to submit a project to the Ministry of Tourism for financial help so as to set up a memorial consisting of a research centre with infrastructure facilities for a library, museum and lecture theatre.
At Kallettinkara of Iringalakkuda, students will be introduced to Indian School of Mathematics in general with stress on the Kerala School along with introductory Sanskrit classes. In future, it is proposed to transform Irinjalakkuda to an academic, spiritual and intellectual tourist destination with varied related activities.
During a recent storm in the neighborhood of the temple and rain, the temple was damaged when a tree fell over it. The trustees of the temple do not have sufficient budget to repair the damage. It will be a good gesture if readers of this article and their friends can help the trustees financially so that the temple can be restored and renovated to its original structure.
*The writer is visiting professor at the International School of Photonics, Cochin University of Science and Technology, Kochi; Kerala University, Thiruvananthapuram, and MG University, Kottayam. He can be reached at nampoori@gmail.com.
