Sunday, 29 December 2024

Muthuswamy Deekshithar - Notes from a 2017 lecture by V Sriram

These are my notes from a lecture at Vani Mahal, Thyagaraya Nagar in December 2017.

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Marvelous lighting at the mini hall in Vani Mahal, such that the speaker Sriram Venkatakrishnan was well lit, without shining on the large the screen displaying his PowerPoint and pictures. There were some projector hiccups halfway but overcome.

From an early move out of Vellore because of a power struggle in the fort, to the expertise of Ramaswami Deekshitar, in Jayadeva's Ashtapathi, to the travels, career and musical virtuosity of his son, Muthuswamy, born by the blessing of the eponymous God in Vaideesvaran temple, it was a comprehensive lecture, as profuse in scholarship as it was delightful in diction and delivery, in not one but three languages.

One could write a twenty page book just from the lecture. I'll list some highlights.

A background of Tanjavur Mahratta politics including Amarasimha, Tulaji, saraboji, Father Schwartz.

Early life in Govindapuram.

Sonti Venkataramana, guru of Thyagayya, was impressed by Deekshitars teaching skills.

Venkatamakhin and his Chatur Dandika Prakasika, containing the melakartas, whose rare copy Muthuswamy had the good fortune of receiving.

Patronage by Manali Muthukrishna Mudali, the Dubash of Madras, and by his son. Songs lauding them, which contrast with other accounts of their lives.

The band in Fort st George, which inspired the nottu swarams, and the adoption of the violin in carnatic music.

Two wives.

A deep knowledge of Tevaram songs, which reflected in several compositions in Sanskrit.

Later settlement in Kanchipuram and encounter with Upanishad Brahmendra, whose mutt survives. Songs on the Somaskanda of Ekamreshvara, with poetic description of the sthala purana, and word play on ma skanda and moola skanda.

A tour of several nearby temples including Kalahasti, Tiruvannamalai, Tirupati. Perhaps a reference to laddu!

A reference to Gnanasambanda as Uttamavipra in his song on Arunachala and his legend of sighting the hill from a distance, and how this episode repeated later in the life of Ramana.

A voyage to Kashi with all its perils, almost surely, entirely by foot. The episode of Ganga gifting a veenai to Baluswamy Deekshitar his brother. Let's just say he came back with a veena, quipped Sriram. A very small veena, perhaps three feet long, which the family preserved and showcased at the Music academy in 1975, the 200th anniversary of Deekshitar.

Life in Tiruvarur, famous also for the life of Nayanmar Sundaramoorthy. The mysteries of the Thyagesa idol, eternally covered from the neck down,which inspired a superb song by another composer. A mysterious closed shrine perhaps of Vishnu behind it, and the oft ignored Valmikinatha, the real moolavar. More on Kamalambal, Katyayani both more celebrated than Neelotpalambal, the primary consort. Katyayani is the Lord's concubine or main Rudra ganika, and the model for all devadasis. In fact it is believe that the devadasi tradition began at this temple. And the erotic sculpture of Uchishta Ganapathi, and it's tantric worship, which Muthuswamy must have practiced as a Sri Vidya upasaka.

The variety and range of his disciples from different communities, including devadasis and the Tanjavur quartet.

Muthuswamy composed music for Rama Ashtapathi, composed by someone else, but unfortunately these haven't survived. Or else we would have known how he set others composition to music, as musician Sriram Parasuram (present at the lecture) observed elsewhere, the speaker adds. (Another singer Ramakrishna Murthy was also in the audience).

Songs on several temples around the Kaveri built including Tayumanavar whom he calls Matrubutha, and various navagraha shrines. Waxing eloquent at Tirukannamangai, where bhkatas believe the devas reside as bees, and the Thayar shrine still has a beehive perhaps two centuries old, that the archaka shows to visitors. Tirumangai Alwar has a mischevous passuram here, where he advises not just Vaishnavas but also Vishnu himself that singing his ten songs will benefit the singer.

Ratnagiri temple, populated by monkeys, where the devotees fill a large copper vessel from the Kaveri, eight km away, and then take it up the thousand steps for abishekam. They did it during his times, as recorded in his song, and they do it today.

Songs on navagraha shrines in the hinterland. Travel to Madurai and Ettayapuram , famous for its betrayal of Kattabommu, but whose king Deekshitar praised. The court appreciated the novelty of the violin as a carnatic instrument. Songs on Meenakshi and Azhagar.

We should recognize that Deekshitar was human, needed patrons, was practical, and not just cover him in saintliness, said Sriram. Final samadhi there. Now, there is a memorial mandapam, which a collector recently wanted to demolish as an illegal structure. Fortunately, it survives.

Originally posted to Facebook on 29 December 2017. Dedicated to Rajagopalan Venkatraman who could not be there, who otherwise would have provided marvelous slide by slide coverage.

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Friday, 13 December 2024

Aryabhata 2025 - ghana citi - Formula for sum of series of cubes

Aryabhata ghana citi
Aryabhata's ghana citi for 2025

Most of us studied not only numbers, but also series of numbers and sums of series in school in mathematics. I am sure most people remember, that the sum of the series from 1 to any number N is given by the formula N*(N+1)/2.

In other words, 1+2+3+4…..+N =  (N * (N+1))  /2

In some school mathematic text books, the name of Carl Friedrich Gauss, the great German genius, is mentioned in association with this series. It seems a German teacher asked his class of nine or ten year olds, what is the sum of the first hundred natural numbers. And Gauss, who was in this class quickly responded, 5050. When asked how we calculated it so quickly, Gauss responded, that he added the smallest and largest numbers 1 and 100, which came to 101; then he added 2 and 99 the second smallest and second largest numbers, which also came to 101; next 3 and 98, then 4 and 97, also each adding to 101. He realized that there were 50 such pairs, each adding to 101, so the sum is 50 times 101 which is 5050. Gauss went on to do amazing things in mathematics, and became one of the greatest mathematical geniuses the world ever saw.

Nice story. Every student and teacher can relate to it. Why don’t we have such stories about Indian mathematicians, except for the famous taxi number 1729 of Ramanujan?

While some formulae in mathematics have names attributed to their inventors or discoverers, there are several mathematical formulae that remain anonymous. All the formula that have names in either physics or mathematics have European or American scientist or mathematician’s names. So, we have Pythagoras theorem, Newton’s formula, Einstein’s formula, Euler identity, and so on. Have you ever wondered why? Why is this Gauss story told without mentioning that Aryabhata gave us this formula for sum of series?

We also learnt that Indians invented zero – we are wrongly told that Aryabhata invented zero. No, Aryabhata did not invent zero. Zero was at least a few hundred years old before Aryabhata was even born. Besides Aryabhata at least Bhaskara is famous as a great mathematician in India. Why do we never learn about some Aryabhata theorem or Bhaskara formula.

Also, even if Aryabhata discovered or invented zero, he must have invented something else also?

Let us discuss one set of things Aryabhata presented, which are given in school textbooks throughout India and the world without mentioning his name. Aryabhata gave not just the formula for the sum of series of numbers, he gave formula for the sum of series of squares and the sum of series of cubes.

In Sanskrit books, the word citi is used for series. Citi  (Sanskrit चिति Tamil சிதி) is literally the word for series of bricks with which a yagna or fire altar for Vedic rituals is made. Aryabhata uses these terms for these formulae

citi: for sum of series of numbers  (1+2+3+4… +N) = N*(N+1)/2

varga citi: for sum of series of squares (1^2 + 2^2 + 3^2+ 4^2 …. + N^2) = N*(N+1)*(2N+1)/6

ghana citi: for sum of series of cubes (1^3 + 2^3 + 3^3+ 4^3 …. + N^3) = (N*(N+1)/2)^2

Varga (वर्ग வர்க) and ghana (घन கன)are the words used in most Indian languages for square and cube. Varga moola and ghana moola are the words used for square root and cube root – incidentally Aryabhata also gave us algorithms to calculate varga moola and ghana moola, but that is a topic for another day.

We learn these formulae in school with Greek notations, invented by European mathematicians in the 18th and 19th century like sigma for sum.

Interestingly the sum of the series of the cubes upto 9, that is, 1^3 + 2^3 + 3^3+ 4^3+…9^3 is equal to 2025, which is the Christian year that comes up shortly. I am sure social media will be full of posters and jpegs and gifs and short videos telling you this interesting fact, and perhaps bated breath narrations of Gauss. And zero mention of Aryabhata. So, here is Aryabhata wishing you a happy 2025.

Ironically Aryabhata knew nothing about this Christian calendar adopted in Constantinople and the Roman empire, a few decades before he was born. He used the Kali Yuga notation in his book on astronomy, giving his own year of birth as 23 years before the 3600th year of the Kali calendar. As 499 AD is Kali year 3600, historians of mathematics believe he was born in 473 AD. In the Kali yuga calendar 2025 is the year 5126 – I am sure enterprising mathematicians will come up with interesting ways to compute this number using Aryabhata’s various formulae.

Related Links

Other essays about Indian Mathematics and Astronomy

My essay in The week magazine on Aryabhata

My essay in Swarajya magazine about Aryabhata

Aryabhata -  CSIR NiScPR Posters