The Genius and the Myth

He ranks with Archimedes, Euclid, Isaac Newton and Leonard Euler as one of the greatest mathematicians of the world. He began a new epoch in Indian astronomy and mathematics, that continued for more than a millenium. His book Aryabhateeyam is a masterpiece of brevity and eloquence.

But what did Aryabhata actually do? Aryabhata did NOT invent zero; or gravity; or the heliocentric system. As I wrote in my first essay, even Indian mathematics and Sanskrit scholars are stunningly ignorant of Aryabhata’s actual accomplishments. Since we are equally ignorant of almost all of ancient India’s glories, this is not specifically galling; just generally abysmal. Only Bhaskara was perhaps as popular and admired, but unlike Newton’s apple or Watt’s tea kettle, or the anecdotes of Birbal or Tenali Raman, we don’t even have popular legends about him. But we are so creative, we blame the British for this situation, decades after they left.

Ever computed a square root? Aryabhata.
Cube root? Aryabhata.
Summed up a series of numbers? Aryabhata.
Series of squares? Aryabhata.
Divided by a fraction by multiplying by its inverse? Aryabhata.
Computed the areas of triangles, circles, trapeziums? Aryabhata.
Calculated sines? Aryabhata. 

And that’s just the simple mathematics we learn in school.

Wait! Did he invent ALL of these? Ah, that’s the question. Aryabhata himself claims not a single invention. He explicitly states that “by the grace of Brahma, the precious jewel of knowledge (jnana-uttama-ratnam) has been extracted from the sea of true and false knowledge (sat-asat-jnaana-samudraat), by the boat of my intellect (sva-mati-navaa).” As Euclid compiled five centuries of geometrical discoveries of the Greeks, Aryabhata compiled several centuries of mathematical and astronomical discoveries of Indians.

Sulba sutra and Jain mathematicians knew how to compute, square roots, but Aryabhata was the first to describe the algorithm. We don’t know if cube roots were calculated earlier, his algorithm is the oldest extant. His sine calculations are considered much superior to those listed by Varahamihira. His kuttakara algorithm to find solutions is considered ingenious even today.

It is not feasible to explain his mathematical and astronomical discoveries in a magazine article for the general reader. There are excellent translations, technical papers, books that do that. This essay’s purpose is to provoke you to read them, and marvel at Aryabhata’s sva-mati-navaa. And to place Aryabhata and his work in historical context.

Manuja Grantham

The eighteen siddhantas were attributed to rishis. But every jyotisha siddhanta after Aryabhata and Varahamihira, is attributed only to men, not rishis. These arose from commenting, understanding, questioning, correcting, improving existing siddhantas and inventing or discovering new concepts. There was no fear or taboo against criticizing a mere manuja like Aryabhata or Bhaskara, rather than a rishi. This era of Mathematics and Astronomy is called “Classical” by historians. I prefer VarahaMihira’s phrase Manuja Grantha.

मुनिविरचितमिदमिति यच्चिरन्तनं साधु न मनुजग्रथितम् ।
तुल्येऽर्थेऽक्षरभेदादमन्त्रके का विशेषोक्तिः ॥१–३॥ – बृहत्संहिता

muni-viracitam-idam-iti yat-cirantanam saadhu na manuja-grathitam
tulye-arthe-akshara-bhedaad-amantrake ko viSheshokti – BrihatSamhita 1-3

 

Translation This (idam) is muni-uttered (muni-viracitam) so sacred (cirantanam) and good (saadhu). Not (na) so manuja-grathitam (man-composed) it is said (iti). If it is not a mantra (amantraka), and meaning (artha) is equal (tulye) but words different(akshara-bhedaa), what’s wrong (vishesha) with it?

Philosophically, this verse by Varahamihira, is as insightful and expressive as Kalidasa’s verse puraanamityeva na saadhu sarvam(Not everything is excellent, simply because it is ancient). 

Aryabhateeyam

The phrase Kusumapure abhyaarcitam gnaanam (knowledge respected in Kusumapura), in Aryabhateeyam hints that he lived in Kusumapura (Pataliputra or Patna). No biography or portrait of any Indian astronomer exists. The pictures of Aryabhata pervading the internet, as well as his statue, are merely artists’ imaginations. Almost all we know about him comes from his books and those of his critics and commentators, like Brahmagupta and Bhaskara I, who mentions Pandurangasvami, Latadeva and Nishanku, as pupils of Arybhata.

He composed:

(1) Aryabhateeyam in 499AD when he was 23 years old. Multiple copies survive in full form.

(2) Aryabhata Siddhanta, which is lost, and known only by quotations from commentators. In this book, Arybhata advocated midnight as the starting hour of each day, instead of sunrise, perhaps based on Surya or Romaka Siddhanta. Aryabhateeyam uses sunrise as day-beginning.

I confine this essay to Aryabhateeyam. It consists of two parts. The first, Dasha Geetika (Ten Songs), lists astronomical constants:

·        Orbital periods and Diameters of Sun, Moon, Planets

·        Number of years in a yuga, yugas in a kalpa, kalpas in a manu

·        Deviation of planets from the ecliptic

·        Epicycles, in different quadrants

·        Table of Sine differences.

 

His first verse is a salutation to Brahma - he was a scientist, but not an atheist. Almost every jyotisha who followed him begins his work with a salutation to his favorite God. Jain mathematician Mahavira begins with an invocation to his namesake, the tirthankara Vardhamana Mahavira. It may also indicate that he was updating the Paitamaha (Brahma) siddhanta, some of whose data, had become obsolete.

The second part, called AryaAshataShatam (i.e The 108 Arya verses) consits of three chapters – Ganita (Mathematics), Kaala Kriyaa (Calculating Time), and Gola (Sphere – i.e. Celestial, Sphere meaning the visible universe).

The siddhantas of later jyotishas were each nearly a thousand verses long. What Aryabhata summaries in one or two verses is explained by them with whole chapters. So cryptic and compact was Aryabhateeyam, it was impossible to understand without bhashyaas (commentaries); such was its impact, that bhaashyaas were written on it centuries after others improved upon his methods. Telugu Marathi and Malayalam commentaries followed those in Sanskrit, Arabic etc; and English translations in the colonial period, which range in appreciation from astonishment to incredulity to calumny.

1.    Ganita - Mathematics

The mathematics set forth by Aryabhata is mostly practical, not theoretical: its primary purpose is astronomy. I mention only simpler concepts in this essay.

It also varies from extremely simple to extremely complex statements, hypotheses, and algorithms.

We must understand that mathematics was not taught to school children, then as it is today; it was perhaps the most advanced of technical subjects and confined to specialists.  Arithmetic symbols familiar to us like + - x ÷ = were only introduced in fifteenth century Europe. Mathematics was not expressed in equations, but in slokas.

Aryabhata gives two line slokas like this:

त्रिभुजस्य फल शरीरं समदलकोटी भुजार्ध संवर्गः

Tribhujasya phala shareeram samadalakoti bhujaardha samvargaH.

 

Bhuja means Arm. Tribhuja means three-armed or Triangle.

Translation “Multiplication (SamvargaH) of perpendicular(Samadalakoti) and half (ardha) the base(Bhuja) results (phala) in Triangle’s (Tribhuja-sya) area(Shareeram).”

A similar verse(sloka) defines the area of a circle as its half-perimeter (or half-circumference) multiplied by its half-diameter (radius) 

This is a simple algorithm, just a formula really, to calculate one value, based on known parameters. A more complex version is his algorithm for summation of a series, which includes several calculations, including for the mean of the series, and encoding an alternate algorithm! This way of stating multiple mathematical formulae is called muktaka by Bhaskara I.

Kaalakriyaa – Time

Aryabhata divided time and circles  with the same geometric units as earlier siddhantas. His major departure, was to define the four yugapadas namely krta, treta, dvaapara and kali, as of equal time; and as the time it took all the nine planets to align, or complete an integral number of revolutions around the earth. He included a biographical note, that 3600 years passed between the beginning of Kali yuga (end of Mahabharata war) and the twenty-third year of his birth. This implies that the constants in DashaGitike were based on his personal observations in that year.

This differed from the smriti definition of the first three yugapadas as four, three and two times as long as the kaliyuga, and offended the orthodox of everyone. Even his followers didn’t accept this division, but they followed his computations and algorithms, as they were significantly better than those of earlier siddhantas.

Gola – Celestial Sphere

Arybahata states that Solar and Lunar eclipses are shadows of the Moon on Earth and Earth on the Moon, respectively. He also stated that the  Sun is the only source of light, and not just planets, but even the stars only reflect sunlight.

Kadamba flower

Aryabhata used the metaphor of a kadamba-pushpa-grantha,  to explain how people and creatures in all parts of the world believe they are standing on top of the world. He introduced another metaphor, for Earth’s rotation: consider a boat-rider on the Ganga, who feels trees on the shore pass him by; whereas, in reality it is the boat that is moving. Similarly Aryabhata suggested, the earth actually rotates, and like trees on a river bank, the stars seem to revolve around it. But it was only a metaphor, not a proof.

He also explains such concepts as Ascencions of the Zodiac, Sine of Ecliptic etc. which are too technical for this essay.

The impact of Aryabhata was phenomenal. Even fervent critics could not ignore him or his works. But he launched an era of manuja grantham, and he was followed by a long line of brilliant scholars, whom we will discuss next.

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This essay was first published as part of a series in Swarajya
For the entire series click this link --> Indian Astronomy and Mathematics   

References

1.      The Aryabhateeyam by Walter Eugene Clark, University of Chicago, 1930.

2.      Aryabhatiyam, translated by KV Sarma and KC Sukla, Indian National Science Academy, New Delhi, 1976.

3.      Facets of Indian Astronomy, KV Sarma, Madras.

 Rishi Siddhaantas and Manuja Siddhaantas

The thousand years before Aryabhata were as rich in intellectual fervour and activity as the thousand years after him. This was the era of the composition of most of the Vedaangas, the creation of such seminal works like Bharata’s NaatyaShaastra, Chanakya’s ArthaShaastra, Vatsyayana’s KaamaSutra, and several magnificent treatises on various subjects. Among these were eighteen jyotisha siddhantas, all attributed to deva-s like Surya or rishi-s like Kashyapa, Atri, Mareechi as described in this sloka.

सूर्यः पितामहो व्यासो वसिष्ठोऽत्रि पराशरः ।
कश्यपो नारदो गर्गो मरीचिर्मनुरङ्गिराः।।
लोमशः पौलिशश्चैव च्यवनो यवनो भृगुः ।
शौनकोऽष्टादशश्चैते ज्योतिःशास्त्र प्रवर्तकः ।।

Surya pitaamaho vyaaso vashishto atri paraasharaH
Kashyapo naarado gargo mareechi-r manu-r angiraaH
lomashaH paulisha-shcaiva chyavano yavano bhrguH
shaunako ashtadasha-shchaite jyoti shaastra pravartakaH

 

This stands in stark contrast with the Siddhantas in the post-Aryabhata classical era, all of which are ascribed to scholarly astronomers, but not rishis. Varahamihira’s phrase manuja-grantham, succinctly describes this.

This was the period during which numerals, the place value system, angular units like degrees, minutes and seconds, trigonometry, and several such mathematical concepts must have been discovered. Instruments like shanku (gnomon), chakra (hoop), gola (armillary sphere), ghati yantra (copper pot) were used.

But all 18 siddhantas are now lost, except the Surya Siddhanta, which was modified and updated in the later centuries. Fortunately, Varahamihira, a contemporary of Aryabhata, wrote a treatise called Pancha Siddhantika, a comparative study of five of these eighteen siddhantas. He quoted and explained several verses from them. So, we understand some concepts of the era.

Types of Jyotisha texts

Jyotisha texts come in several categories. Siddanta-s are once in a century grand texts, composed by superlative scholars. A siddhanta may have several commentaries, called bhashya-s, in the succeeding centuries. For practical use, more compact books called karana-s were composed, which was used by pandits to prepare almanacs/calendars called panchaanga-s for public use. The latter tradition is still extant.

It is my belief that the various texts on astronomy and mathematics rival the commentaries and compositions on the Ramayana and Mahabharata. So rich and so widespread was the literature.

Pancha Siddhantika

The five siddhantas Varahamihira studied, those of Pitamaha (Brahma),  Vashishta, Surya, Romaka and Paulisha, explain motions of planets (in a geocentric model), prediction of eclipses, sine tables, celestial longitudes and latitudes. None of these are mentioned in Vedanga Jyotisha. They vary mostly in minor details, which Varahamihira explains. The small Vedic yuga of five years was dropped, and the humongous yuga of 432000 years used. We have no idea when or how this changed. A day count, ahargana, counting number of solar days (regardless of month or year) since the start of the Kaliyuga, which began when the Mahabharata war ended, came into vogue. Kaliyuga years are found inscribed in several royal inscriptions; for example, the Anamalai inscription of Maranjadayan Varaguna Pandyan in Madurai.

The solar zodiac is used extensively. It was most probably borrowed from the Greeks or Babylonians. The solar zodiac is a popualar theme on ceiling sculptures of temples in Tamilnadu, like this one in Kudumiyan Malai, Pudukottai.

Romaka (also called Lomasha) and Yavana refer to a Roman and a Greek, Paulisha to a Paulus Alexandrinus, say historians of science. While some foreign ideas were obviously borrowed, there is a puzzling absence of inclusion of other ideas, including those of Euclid, Ptolemy, or Archimedes. Whereas the Greeks developed an epicyclic theory of planetary motion, Indians developed a theory based on air strings pulling the planets. Geometrically, these are simply different epicyclic model than those used by the Greeks. They involved extremely complicated geometry, trigonometry and algebra, but they were quite accurate in predicting eclipses, solar and lunar, the biggest challenges of Indian astronomy.

That Mercury and Venus had a different type of orbital movement, from the other planets, Mars Jupiter and Saturn, was realized. Siddhantas explain eight types of planetary movement.

A vocabulary of scientific and technical words developed, to describe both such astronomical concepts and mathematical ideas and theorems.

From the earlier knowledge of hypotenuses and circles, as found in Sulba Sutras, we can understand that the concepts of sine, cosine and other trigonometric ideas arose. The Indian sine was not the opposite/hypotenuse that we learn in school today, but the radial sine (abbreviated as R-sine), called the ardha-jyaa (half-bowstring). A chord connecting the ends of an arc looks like a bow (Sanskrit: chaapa or dhanush). When seen as part of a circle, the radius of the circle (CM )is the hypotenuse of the triangle (CMA) formed by the half-chord (MA), the radius touching the top (M) of this chord, and the segment (CA) of the radius dividing the chord into two equal halves. In Indian siddhantas, in the table of sines, expressed as a series, only the numerators are listed. Hence they are radial sines (multiplied by radius). The word for cosine is koti-jyaa.

The word jyaa and this concept of trigonometry traveled from India to Baghdad in the eighth century during the reign of Caliph al-Mamun, along with the zero, the decimal place value system, Indian numerals (now called Arabic numerals) and the works of Aryabhata and Brahmagupta. It transformed into the Arabic word jyaab or jeyb which means pocket. This then was taken to Europe by Leonardo Fibonacci, an Italian merchant, in the twelfth century, and translated into Latin as sinus, and later into English as sine. Then it came to India under English colonialism, making a full circle (pardon the pun) into our mathematical textbooks as sine. We learn trigonometry as the gift of the Europeans, not realizing its Indian origin.

Angular measurements called kalaa (degrees) liptaa (minutes) and viliptaa (seconds), were used, based on the sexagesimal system (Base-60) rather than decimal, which hints at a Babylonian origin. In addition, a sub division of the second into sixty parts and division of the cirlce into twelve parts (called raashi) also existed. Angles were often represented in karana texts with five aspects, not just the three we use today.

The division of time was also sexagesimal, with a day consisting of sixty naadis, each naadi of sixty vinaadis. Remember, the naadi existed in the Vedic period; was it indigienous or imported? It’s not one of several mysteries.

Step by step mathematical procedures (now called algorithms, after the Uzbek mathematician, Mohammad ibn Musa al-Khwarezmi) also emerged in the era of 18 Siddhantas. The place value system and zero were invaluable in developing algorithms for multiplication and division, square and cube roots, and several algebraic procedures solving indeterminate linear equations.

Ujjain Meridian

Two millennia before the world adopted the Greenwich meridian, Indian astronomers used the Ujjain meridian, as the prime meridian of longitude in India. This is the longitude that passed from north pole (Meru) to south pole (Vadavamukha). That the earth was a globe, not a flat plain was well understood by astronomers. They believed that Devas lived at Meru and Asuras at Vadavamukha, and Mankind in between.

गगनमुपैति शिखिशिखा क्षिप्तमपि क्षितिमुपैति गुरु किञ्चित् ।
यद्वदिह मानवानामसुराणां तद्वदेवाधः ॥ १३– ४ ॥  Pancha Siddhantika 13-4

Gaganam-upaiti shikhi-shikaa kshiptam-api kshitim-upaiti guru kincit
Yadvad-iha maanavaanaam-asuraaNaam tadvadeva-adaH 

The flame (shikhaa) of a lamp(shikhi) points skywards (gaganam) and a heavy (guru) object (kincit) thrown (kshiptam) skywards falls back to earth (kshiti); this happens in the lands of men (maanavaanaam) and asuras (asuraaNaam)

This was one concept of gravity, before Newton changed it.

उदयो यो लङ्कायां सोऽस्तमयः सवितुरेव सिद्धपुरे ।
मध्याह्नो यमकोट्यां रोमक विषयेऽर्धरात्रं स॥ Pancha Siddhantika 15-23

Udayo yo lankaayaam sa-astamaya savitur-eva siddhapure
Madhyaahno yamakotyaam romaka-vishaye arddha-raatram saH

Translation When it is Sunrise (udaya) in Lanka, it is Sunset (astamaya) in Siddhapura, Noon (madhyaahna) at Yamakoti,  Midnight (arddha-raatra) in Romaka-vishaya 

Lanka is not the Sri Lanka we know, but the point where the Ujjain meridian intersects the equator. Ujjain was a major center of learning in ancient India, and is also perhaps closes to the Tropic of Cancer (Karkata). We don’t know what places Yamakoti and Siddhapura signify, perhaps they are also place marker names like the equatorial Lanka.

While all the other Siddhantas determine time with Ujjain as the prime meridian, Romaka Siddhanta says the days starts with sunset at Yavanapura, which is not Athens or Rome, but Alexandria in Egypt.

The logical thought process which inspired the use of Ujjain and Lanka for calculations is simple, but brilliant. Longitude and latitude determine local time. So, the times of sunrise, sunset, moonrise, eclipses, will vary from place to place. Once the calculations are made for a prime meridian like Ujjain, local panchaangam-s can be prepared with only minor changes applied for local longitude and latitude – these are called deshantara, Each Siddhanta has a section about it. 

Celestial longitudes and latitudes were easier to calculate, than those on earth. The Surya Siddhanta lists Rohitaka (Rohtak, Haryana) and Kurukshetra and other cities on the Ujjain meridian. Others list such places as Kanyakumari, Malavanagar, Sthaneshvar, Vatsyagulma, Mahishmati, Vananagara as cities on the Ujjain meridian.

Some  trivia : Ujjain passed on its torch to Madras, briefly. Today, Indian standard time is set on longitude 82.5E,  based on Greenwich meridian. But for about a century, the Madras meridian was used as the prime meridian, especially for railway clocks.

For the entire series click this link --> Indian Astronomy and Mathematics   

References

1.      Surya Siddhanta, by Phanindralal Gangooly

2.      Pancha Siddhantika, edited by KV Sarma

3.      Pancha Siddhantika, edited by G Thibaut, Sudhakara Dwivedi

The Vedaangas are six subjects created to assist the study of the Vedas. Four of the Vedaangas are about linguistics; the other two have significant mathematical sections.

Mathematical patterns in Akshara, Vyaakarana and Chandas

As we saw in chapter three, the Sanskrit alphabet is a masterpiece of analysis and organization  - and this may predate its written form. The sounds are classified as svara (vowels) and vyanjana (consonants), then grouped by origin in the mouth, voicing and aspiration. This may seem unremarkable, until we learn a language which practically don’t use vowels, like Hebrew or Arabic. Pertinent to mathematics, though is the differentiation of vowels as short (hrsva) and long (dheerga) based on duration of pronunciation, called maatra (measure). This is standardized in every Indian literary language, including the Dravidian family, but missing from European languages. Even after hearing every name, city, object creatively uttered by foreigners, rarely do we appreciate this. Ah, for an Oxbridge accent!

There is a separate classification of syllables, rather than letters, as laghu (light) or guru (heavy), which is the basis of generation and classification of chandas (prosody). Each poetic meter has a different number of syllables.  The binary nature of the syllables, and the various possible combinations for letters led to the development of combinatorics; Meru-prastara, a precursor to Pascal’s triangle; and algorithms to find metrical patterns or their various aspects based on the number of syllables.

The most common chandas or metre called anushTubh has four paadams ( quarters) of eight syllables each.

A longer meter called mandakraanta has four quarters of seventeen syllables each.

For example, consider a three syllable meter. There are eight possible combinations of guru (G) and laghu (L) syllables.

Table Prastara for three syllables

Pingala provides six different pratyayas (mathematical procedures or algorithms), summarized in this table.

Is this mathematics or linguistics, one might ask? Even the linguistics texts have a mathematical structure.

Sanskrit literature is in one of three forms: chandas (verse), champu (text) or sutra (brief text). All the Vedangas, composed in the Vedic era, except Jyotisha are in sutra form, in very cryptic notation, akin to modern mathematical notation. Ironically all the ganita siddhantas of post Vedic period are in chandas, poetic form. What an irony: Grammar and poetry books in mathematical notation, but mathematics books in verse!

In fact, we find the oldest mentions of zero, Shunyam and Lopah, in Panini’s Grammar Ashtadhyaayi and Pingala’s prosody ChandaSutra, not in the Jyotisha text!

Panini’s Ashtadhyayi

Long before Panini, Sanskrit had eleven books on grammar, by Indra, Galava, Gautama etc.,whom Panini himself mentioned. The Siva Sutra, a brilliant arrangement of letters into 14 groups, in algebraic notation, also pre-dates Panini and was fully exploited by him. Panini’s grammar is so algebraic in its notation, it has been an inspiration for later mathematicians, say experts. This was centuries before algebra developed! His grammar is generative rather than an analytical or descriptive grammar. No other language in the world had such a grammar book, until  mathematicians developed similar notations for computer languages in the 20^th century.

Let us look at one sample – sandhi rules for combination of letters and sounds.

First if you look at English words, what rules govern spelling when words combine?

Work + ing  =  working                           fly+able   =  flyable

Bowl  + ing  =  bowling                            ply+able   = pliable

use  +  ing    =  using                               note+able = notable

bat   + ing    =  batting                             hit+able   = hittable

run  + ing    =   running                           free+able = freeable

 

In some cases, two words simply merge together, with no change to either. In some cases, as in hittable and running, the last letter of the first word doubles. In some cases, as in using and notable, the last letter of the first word is dropped. In some cases as in pliable the last letter of the first is transformed from y to i.

There is no English grammar book with such formal rules as is seen in the books of Sanskrit grammar. English spelling and pronunciation have undergone drastic changes over the last few centuries, unlike Sanskrit whose rules of grammar havent changed much in millennia. But most people accept these rules at least in an informal way.

Sanskrit, though has very specific rules for how letters merge, when letters drop, add or transform

 

Examples

veda + anga   =  vedaanga
kamalaa + ambaa    =  kamalaamba
simha + aasanam = simhaasanam
padmaa + aasani = padmaasani

Examples

raaja + indra   =  raajEndra
chola + eeshvara   =  cholEshvara
loka + eka      =  lokAIka
purusha + uttama = purushOttama
kula + uttunga = kulOttunga

SivaSutras also called Maheshvara Sutras, is a set of shlokas, that organized the Sanskrit alphabet into 14 subsets. Each subset consists of some letters, not in the original alphabetical order, and a terminal letter. A two letter notation, the first of which is the a letter that indicates the first letter of the subset, te second of which is a terminal letter, is used to indicate any desired subset.

अइउण्	a i uN		Examples
ऋऌक्	R L k
एओङ्	E O ng		अक्  a k	अ इ उ  ऋ ऌ   a i u R L
ऐऔच्	ai au c		यण्  ya N	य व र ल ya va ra la
हयवरट्	ha ya va ra T		तय् ta y	त क प ta ka pa
लण्	laN
ञमङणनम्	nya ma nga Na na m		अच्  ac	All vowels
झभञ्	jha bha ny		हल् hal	All consonants
घढधष्	gha Dha dha SH
जबगडदश्	ja ba ga Da da sh
खफछठथचटतव्	kha pha cha Tha tha caTata v
कपय्	ka pa y
शषसर्	Sha1 SHa2 sa r
हल्	ha l

SivaSutras

Panini and Backus-Naur Form

In the 1950s, IBM Corp developed the ForTran computer programming language, the first high-level language for computer programming. The team’s leader John Backus developed a notation called Backus-Normal Form, later altered to Backus-Naur Form, to notate grammar for ForTran. 

Peter Zilahy Ingerman, Manager of Langauge Systems Standards and Research, Radio Corporation of America, in 1967, wrote a letter to the journal Communications of the ACM, suggesting that this notation be renamed Panini-Backus Form. He opined that, “Panini had invented a notation, which is equivalent in power to that of Backus and has many similar properties.”

Linguists since then have hotly debated the virtues and pitfalls of Panini’s system. But it is most proabable that this is the origin of the rumour that Sanskrit is the most suitable language for computer programming. That is not true; but surely, Panini’s grammatical notation has the greatest similarity to the grammatical notation of computer languages.

Sulba sutras

Egyptians and Greeks developed geometry for earth (geo) measurement (metry). We may say Indians developed vedi-metry: the sulba sutras are manuals for configuring vedis(altars) of different shapes for performing yajnas(Vedic sacrifices). They were measured using ropes (rajju or sulba). Baudhayana, Kaatyaayana, Apastamba, Maanava, Maitraayana, Vaaraaha, Vaadhula are seven Sulbasutras named after their composers, that survive. Their contents are very similar.

विहारयोगान्व्याख्यास्यामः । Aapastambha Sulbasutra 1.1
Vihaara-yoga-vyaakhyaasyaamaH.

Translation We make known (vyaakhyaasyaama) constructing figures (vihaara-yoga) 1.1 

रज्जुसमासं वक्ष्यामः । Kaatyaayana Sulbasutra 1.1
Rajju-samaasam vakshyaaamaH.

Translation We explain (vakshyaama) by combination (samaasam) of ropes (rajju)

Unlike axiomatic Greek geometry, which is very theoretical, Sulbasutras are practical or applied. They specify rules and methods for construction of vedi-s, finding true east, calculating areas, diagonals, choosing clay, making bricks, calculating number of bricks, etc.

Praachi – Finding true east

समे शङ्कु निखाय शङ्कुसम्मितया रज्जवा मण्डलं परिलिख्य यत्र लेखयोः शङ्क्वग्रच्छाया निपतति तत्र शङ्कू निहन्ति सा प्राची ||  कात्यायन शुल्बसूत्र  1–2

Translation

Plant a shanku (gnomon), on level ground.
Draw a circle with rope measured by shanku.
Where shadows fall on circle, fix a stick (morning, evening)
That line is the praachi (line connecting the two sticks), the east west line

As the concept of ayana was well understood, this explains that true east is not merely the direction the sun rises, but its midpoint over the full year. The method of finding true east, using simple devices like a stick, rope, pegs is very typical of Indian astronomy.

Diagonal of a square

चतुरश्रयाक्ष्णयारज्जुर्द्विस्तावतीं भूमीं करोति । समस्य द्विकरणी ॥ Aapastambha 1.5

chaturashraya akshnayaa rajju dvis-taavatim bhoomi karoti । samasya dvikarani  ॥

Translation The rope (rajju) on the diagonal (akshnayaa) of a square (chatur-ashra) produces (karoti) double (dvi)  the area (bhumi) of the square. It equals (sama) √2 (dvikaraNi) of the side of the square.

Dvikarani literally means double-maker. To construct a square twice the size of a given square, the latter’s diagonal should be used as the side for the bigger square. The word karani means square root, diagonal of a square, producer etc.

Square root of two

प्रमाणं तृतीयेन वर्धयेत्तच्च चतुथेनात्मचतुस्त्रिंशोनेन सविशेषः ॥ Aapastambha 1.6

pramaaNam truteeyena vardhayet-tacca caturthena-aatma-catus-trimsha-unena sa-visheshaH ||

Translation The measure (pramaaNam) is to be increased by  its third (truteeyena) and this (aatma) again by its own fourth (caturthena) less (unena) the thirty-fourth part(catus-trimsha) approximately (sa-visheshaH) 1.6

√2 =  1 +    1    +    1    ( 1 -    1   )

               3        3*4           34

 

This is an excellent approximation; but also an indication that they didn’t understand the irrational numbers. visheshaH means special; here the special nature is interpreted as the approximate nature of the result. A similar sutra exists for square root of three, which demonstrates the general concept.

Squaring a Circle

मण्डलं चतुरश्रं चिकीर्षन् विष्कम्भं पञ्चदशभागान्कृत्वा द्वावुद्धरेत् । त्रयोदशवशिष्यन्ते । सानित्या चतुरश्रं ॥ 3.3 ॥

Translation Mandalam catur-ashram cikirshan vishkambham panca-dasha-bhaagaan-krtvaa dvaar-uddharet | trayo-dasha-avavashishyante | sanityaa catur-ashram ||

To transform a circle (mandalam) into a square(catur-ashram), the diameter(vishkambham) is divided into fifteen parts (pancha-dasha-bhaagam) and two (dvaa) of them are removed(uddharet), leaving (avashishyante) thirteen parts (trayaodasha). This gives the approximate (side of the desired) square.

But this shows not only an understanding of equation famously called Pythagoras theorem, but also an understanding that it applies not only to whole numbers. As Baudhayaana is oldest Sulbasutra with this sutra, Fields medalist Manjul Bhargava is among many mathematicians who thinks Pythagoras theorem should be renamed after Baudhaayana. 

In the Classical era, Aryabhata onwards, the phrase bhuja-koti-karna nyaaya denotes this relationship of hypotense to a right triangle, or diagonal to a square, with bhuja, koti and karna meaning, base, perpendicular and  hypotenuse. The SulbaSutra words for these are paarshva-maani, paryanka-maani and akshnayaa. Such a vast period of time elapsed between the Sulba sutras and Aryabhata, that even the scientific vocabulary transformed.

Vedanga jyotisham

I explained some of the ideas in the Rg and Yajur Vedanga Jyotishas in the previous essay. Ill touch upon some other concepts here. One was the categorization of some of the twenty seven stars into fierce, and cruel; this is not found in later siddhantas. A sloka referncing each star by a single letter name is another unique, unrepeated highlight.

Another is the use water clocks. Pots of different sizes, called कुडव, आढक, द्रोण  kuDava, aaDhaka, drona were used. They represented the time consumed by the complete draining of water in a pot. 16 kuDava-s is a aaDhaka, 4 aaDhakas is a drona. A naadi is the time it took for 45 kuDavas of water to drain away. No explanation is given; these must have been taken from common usage.

Other time measures like kala, muhurtha, parva, lagna are introduced. Most of the slokas in Vedanga Jyotisha are procedures for calculating various time based phenonmenon. These include calculating the equinoxes, increase in day and night lenghts, differences between lunar and solar days etc. They are too complicated for this essay, so I will omit them. Remember the purpose of the subject was explained as kaala-vidhaana-shaastram, the science of calculating time.

The simple mathematical calculation of proportions called thrai-raashika, (Rule of Three) is seen here. If x tasks take y of units time, how many units of time does z tasks take, is an example. Given any three of these, the fourth can be calculated. This arithmetic, perhaps the simplest algorithm for calculating an unknown based on known quantities, which we learn in our primary schools, is as old as the Vedangas.

The astronomical phenomenon not discussed are eclipses, their calculation or prediction, celestial longitudes and latitudes, orbits of planets etc. These subjects were studied in the subsequent era of the 18 Siddhantas. 

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References

 Pre-Siddhantic Astronomy, Lecture by Prof RN Iyengar, Seminar on Contributions of KV Sarma, KV Sarma Library Chennai, 2018

NPTEL Lectures on Indian Mathematics by Profs MD Srinivas, MS Sriram, K Ramasubramanian

Boks

Apastambha, Baudhayana Sulba sutras

Vedanga Jyotisha, TS Kuppana Sastry, edited by KV Sarma, 1984

Vedic Mathematics is a more popular catchphrase than Vedic Astronomy. The adjective Vedic has become associated with anything Sanskrit! The four Vedas, though, are primarily slokas about ritual, prayer, philosophy. But numbers are mentioned, even revered; 28 stars, new and full moon, eclipses seasons, days, periods, years, etc are mentioned. From these we can glean some facts and deduct concepts.

All arts and sciences, of whatever variety that existed, had their own separate texts. Invariably, older texts or anthologies were discarded or forgotten over time, because newer compositions replaced archaic knowledge. Panini’s grammar, for example, made eleven previous Sanskrit grammars obsolete. Even scripts of written languages underwent this change, as Brahmi, Kharoshti inscriptions in India and Sumerian Cuneiform and Egyptian heiroglyphs show us. As an aside: we rarely hear of Sumerian commercial mathematics or Pharaonic mathematics.

We don’t know what other texts or anthologies, existed during the Vedic era. Also, the Vedas were transmitted from one generation to the next entirely orally. If any other books were written or even transmitted orally, they are mostly lost to us. The Vedas, alone, because they were considered sacred, were passed on thousands of years later, unmodified. No Indian astronomer, from Aryabhata in 5^th century to Pathani Chandrasekar in 19^th century refers to the Vedas, as a source of their astronomy or mathematics – they only refer to jyotisha siddhantas of earlier astronomers. Nevertheless some historical information may be gleaned from the Vedas.

Remember also, the Vedas are a compilation of various slokas of rishis – we don’t truly know the period of their composition. The puranas, and Vedic commentaries, attribute yugas, millions of years to history. Geology, archaeology, biology, anthropology consider Indian civilization to be a few thousand years old; how many thousands is subjective. Since it is least controversial, I call the period before the 5^th century BC as Vedic, without an upper bound.

Divisions of Day

The concept of solar (savana or divasa), lunar(tithi), stellar(nakshatra) days co-existed, as we saw earlier. 28 Nakshatras are listed, but Abhijit was dropped, with no loss to astronomy. The 12 zodiac series from Mesha (Aries) to Mina (Pisces) was a later import, probably from Babylonia.

The day was divided into two major parts: poorvaahna (forenoon) and aparaahna(afternoon). Poorvahna itself divided into two parts, praatha and samgava. Aparaahna into three parts, madhyaahna, aparaahna and saayaahna. Even today, sandhyaavandanam and maadhyaahnam are important Vedic rituals, performed by millions daily.

This five part division was at some point superceded by a division of the day into thirty muhurtha-s, fifteen each for daytime and nighttime. Today we number the hours, from 1 to 12 or 1 to 24 (this is an import from Egyptian astronomy!) But Vedic muhurthas were named, not numbered. Much later, during the 18 Siddhantas era, and naadi-s supplemented muhurta-s, as we saw in Aryabhata’s sloka. A naadi equals two muhurta-s.

We use numbered tithis : prathama, dvitiya, trithiya, chaturthi, panchami (first, second, third, fourth, fifth) upto trayaodashi, chaturdashi. But the Vedas had names, not numbers, for each tithi. In fact the tithi-s had different names for the day versus nighttime, and different names during the waxing (shukla paksha) versus waning (krishna paksha) phase. So there were sixty tithi names for a lunar month. Similarly there were sixty muhurthas. All these are named in Taittreya Brahmana (3.10). A sample of these names are shown in the accompanying tables.

Divisions of the Year
Just as the day was divided into parts, the year also divided into days(Dina), months(Maasa), seasons(Rtu), Ayana(half-years). The months were named Madhu, Madhava, Shukra etc, not the Chaitra, Vaishaka, series we use today. Each rtu is a season of two months; India experiences six seasons, not four like Europe. The ayana is the apparent traversal of the Sun between northern and southern latitudes; uttaraayana (also called devaayana) is the traversal from south to north; dakshinaayana (also called pitraayana) is traversal from north to south.

Disparity threatened; the months were lunar; 12 months came to 354 days; but the solar year was eleven days longer. To resolve this disparity which threw off the calendar, the concept of adhika maasa was introduced. Adhika means extra. Basically, it was a leap month, similar to a leap day.

A five year cycle after which five solar years and 62 lunar months aligned again, was called a yuga (which means conjunction).

These demonstrate that the concept of time evolved even during the Vedic period.

When the yuga transformed to a tremendous period of 4,32,000 years and a chaturyuga ten times as long is unclear; this is the period the puranas mention. The five year yuga is not even referred to by astronomers of the Siddhanta or Classical period, except Brahmagupta, who rejects it, and the entire Vedanga Jyotisha, without explanation. A fine historical and philosophical anomaly - the very Brahmagupta who is dismissed as orthodox for rejecting Aryabhata’s alleged heresies, is not himself considered a heretic for rejecting an entire Vedanga!

The invocatory sloka of Vedanga Jyotisha mentions all these divisions of the year, saluting Prajapati, the Creator, as one whose limbs are days, seasons, months etc.

पञ्चसंवत्सरमयं  युगाद्यक्षम् प्रजापतिम् ।
दिनर्तोयनमासाङ्गं प्रणम्य शिरसा शुचिः ॥

Pancha-samvatsam-ayam yugaadi-aksham prajaapatim
Dina-rtu-ayana-maasa-angam praNamya shirasaa shuchi

 

This is a common theme throughout the Vedas and Hindu philosophy: not only the Sun, Moon and planets, besides the Earth are considered manifestations of divinity, all celestial phenomena described as parts of their divine bodies. Even numbers are considered divine, and offered salutations. Everything was sacred, which rails against our perception of science today, when considering anything sacred is unscientific.

Numbers

The Taittiriya Aranyakam 4.69 of the Krishna Yajur Veda has this sloka

सकृते अग्ने नमः । द्विस्ते नमः। त्रिस्ते नमः। दशकृत्वस्ते नमः । शतकृत्वस्ते नमः । आसहस्रकृत्वस्ते नमः ।अपरिमितकृत्वस्ते नमः । 

Sakrte agne namaH. Dvis-te namaH. Tris-te namaH. Dasha-krtvas-te namaH. Shata-krtvas-te namaH. Aa-sahasra-krtvas-te namaH. Aparimita-krtvas-te namaH. 

Translation NamaH (salutations) to Agni. NamaH twice. Namah thrice. Ten times namaH. Hundred times namaH. Thousand times namaH. Unlimited times namaH.

The Taittiriya Samhita 7.2.20 has this sloka

शताय स्वाहा सहस्राय स्वाहा अयुताय स्वाहा नियुताय स्वाहा प्रयुताय स्वाहा अर्बुदाय स्वाहा न्यर्बुदाय स्वाहा समुद्राय स्वाहा…… परार्धाय स्वाहा

Shataaya svaahaa sahasraaya svaahaa ayutaaya svaahaa niyutaaya svaahaa prayutaaya svaahaa arbudaaya svaahaa nyarbudaaya svaahaa samudraaya svaahaa … paraardhaaya svaahaa

Translation Svaahaa to a hundred, thousand, ten thousand, lakh, etc. up to one trillion.

This prayer or offering follows a series of prayers, to the first twenty natural numbers, odd numbers, even numbers, multiples of four, five, ten, twenty, forty and fifty. The religious significance is not my scope. What they counted a trillion of, is an enigma. But these slokas demonstrate the use and understanding of decimal (base 10) numbers and large numbers. 

Note the absence of the familiar laksha (lakh) and koti (crore). By the classical era, new names for some larger numbers emerged. Aryabhata, for example, uses koti for arbuda, and vrndam for samudra.

The fractions paada, shapha, kushTha and kalaa respectively 1/4, 1/8, 1/12 and 1/16 are mentioned in the Vedas, too.

These are all names of numbers only, not mathematical operations or algorithms, which are explained in Vedangas.

Other stars, planets, constellations

Five planets are mentioned, but only Brhaspati by name, and Venus as Vena rather than Shukra. The most famous constellation was Sapta Rishi, (a section of Ursa Minor), stars in the polar regions, which revolved around Dhruva, the Pole star. There is a beatiful poem, comparing stars rising after Vrkam(Sirius), as birds of heaven chasing the wolf.

The brightest star of the southern hemisphere suddenly became visible during this era. This star was called Agastya (Canopus) and linked with the story of rishi who headed south, requesting the growing Vindhyas to stop until he returned. Perhaps the legend captures the period of a tilt in the earth’s axis, when the night sky at that latitude became on Aryavrata. Another southern star is called Vishvamitra, and three surrounding stars, Trishanku.

Comets, Meteors, Eclipses

A prayer in Atharva veda mentions comets (ulkaani) and meteors (dhoomaketu). Strangely, except for Varahamihira, no classical astronomer mentioned these objects.

अत्रि: सूर्यस्य दिवि चक्षुर आधत् स्वर्भानोर अप माया  अगुक्षत ||
यं वै सूर्यं स्वर्भानुस् तमसा विध्यद आसुर: |
अत्रयस् तं अन्वविन्धन् न हि अन्ये अशक्नुवन् || Rg Veda 5.40.8-9 

This sloka refers to the rishi Atri “who set the press stone, revered the Gods, dispelled the illusions of Svarbhanu and restored the Sun’s light”, a reference to a solar eclipse. “Atri and his sons alone could restore the Sun when Svarbhanu had covered him with darkness”, it continues. By the era of the 18 Siddhantas, Indian astronomers understood eclipses, and could predict them. We will come to that after a look at the Vedangas.

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References

Facets of Indian Astronomy, by KV Sarma
Vedanga Jyotisha, TS Kuppana Sastry, edited by KV Sarma, 1984
NPTEL Lectures on Indian Mathematics by Profs MD Srinivas, MS Sriram, K Ramasubramanian

A joke doing the rounds on whatsapp goes, Aryabhata asked his wife what the chances were of his being allowed to stay out past midnight on New Year’s eve, and that’s when he discovered zero. There is a double whammy of breath-taking ignorance in this joke. Aryabhata didn’t invent zero. The zero was in use in India, at least seven hundred years before Aryabhata. This is roughly like making a joke with a punchline involving Edison inventing the telescope.

The other irony, is that Aryabhata was the first Indian astronomer to propose using midnight to mark the beginning of a day. Until then, sunrise marked the beginning of a day. The whatsapp joke is therefore, historically flawed on two counts. Isnt it ironic, that Indians don’t know what their most famous mathematician Aryabhata discovered?

Mathematicians Anonymous

In the Vedanga book on prosody Chanda Sutra, Pingala uses the word shunyam for zero. In the Vedanga book on grammar Ashtaadhyaayi, Panini uses the word lopaH, which some historians interpret as another word for zero. So strangely, it is not in the Jyotisha or Sulba Sutras, which deal with mathematics, but in the Vedanga books on language, that we first encounter the word. Neither Panini nor Pingala claim to have discovered zero, or bother to explain the concept. They just assume that the reader understands.

So the zero was quite well known by their era, estimated to be the second century BC or sixth century BC. Some anonymous mathematician gave us the zero.

We learn numerals, that is, the symbols 0,1,2,3 upto 9 and assume that that is always how numbers have been written. But the surviving literature shows us a slow evolution, with several gaps in the historical record. We also assume that decimal numbering system (base 10) is normal, as the whole world uses it today, but as we will see, various civilizations developed different number bases and strikingly strange systems for notation of numerals.

Not only the zero, but also the place value system of decimal numbers is one of India’s greatest contributions to the mathematics of the world. Before diving into various of aspects of mathematics discovered and used in India, either by astronomers or others, it will be useful to understand how numbers were named and notated.

Numbers in Words

We can write any number, for example, 1729, in numerals; or in words as one thousand seven hundred and twenty nine. Almost all literate societies developed words for large numbers. Indians perhaps came up with the largest names and numbers. All Indian languages have adopted several hundreds, perhaps thousands of Sanskrit words; including words for numbers, sometimes in modified form. Ek do theen, a song from 1980s Hindi film Tezaab, perhaps taught more non-Hindi speaking Indians to count up to 13 than most government funded schools. Ek do theen and all Hindi number names (in fact, number names in the Indo-Aryan langauges) come from the Sanskrit original Eka, dvi, threeni, chatur, pancha, shad, sapta, ashta, nava and dasha. The southern Dravidian family of languages, though, use a different set of number names (one ancient distinction from the Aryan family). The Tamil numbers from one to ten are onru, irandu, moonru, naangu, ainthu, aaru, ezhu, ettu, onpathu and patthu. Malayalam uses the same names, but Kannada, Tulu and Telugu use a set of words similar to Tamil, with common roots.

Names of higher numbers exhibit an interesting feature. In English, and other Indo-European languages, we write and say the most significant numeral first, and proceed in descending order to the least significant value: Thousands, hundreds, tens, ones. The Dravidian languages all follow this sequence, though unrelated to the Indo-European family. So 1729 is aayiraththu (thousand) ezhu (seven) nootru (hundred) irupatthu (twenty) onpathu (nine). But Sanskrit number names go from least significant to most, in ascending order. Fourteen is chatur-dashi (four and ten); twenty-four is chatur-vimshati (four and twenty); and one thousand and eight is ashta-sahasram (eight and thousand).

Even as numeral notation, the notation was from least significant digit (LSD) to most significant digit (MSD). So 1729 was written as 9271 (and would be called nava-vimshati sapta-shata sahasram).

We only find numbers written as words in ancient texts, not just Vedas and Vedangas, but also other literature, scientific or artistic. Numeral notation had as anonymous an inventor, as zero and the place value system (or any of the oldest concepts).

But remarkably two other methods of expressing numbers in words were also developed.

BhutaSankhya भूतसङ्खया

Bhuta means body. Sankhya means number. In the BhutaSankhya system, instead of the familiar number names, the names of objects representing their values are used. The table below features, Numbers 0 through 9, their familiar number-names, the BhutaSankhya synonyms and the meanings of those words in English.

Using this system, the number 352 could be expressed as loka-paandava-netram or guna-bhuta-hasta. 1729 would be chandra-svara-ashvini-ratnam. They also would mix up the familiar number names with the Bhutasankyha synonyms. For example, 5482 could be written as pancha-veda-ashta-netram. (For convenience of readability, I use the MSD to LSD order; in actual Sanskrit texts, it would be the reverse). These BhutaSankhya phrases make no sense as grammatical or semantic constructs, but they have the advantage of being expressed in words. As all our Sanskrit texts including siddhantas on astronomy and mathematics were composed in verse, they only have to fit the syllabic rules of chandas. This also had the advantage of writing very large numbers without mentioning all the place names like. For example, 1,44,63,895 could be written as eka-veda-veda-rasa-guna-vasu-ratna-paandava without all the mentions of crores, lakhs, thousands, hundreds etc. As compact as possible using words alone.

Obviously, some of the words are based on familiar physical objects. We have two eyes or lips, there is only one moon, eight directions, seven musical notes and nine precious stones, and so on. Others are based on cultural artifacts of India – the three lokas, four vedas, seven rshis and mountains, six seasons (rtu), and even three Ramas (Parashu Rama, Rama of Ayodhya, Yadava Balarama) and the three eyes of Siva.

This is not confined to single digit numbers. Jina or arhat (for tirthankara) stands for 24, avatara or anguli stands for 10, nakshatra for 27, deva for 33 etc. So the number 332724 may be written as deva-nakshatra-jina, wherein three words represent six digits.

Katapayaadi कटपयादि

The other system to represent numbers was using letters instead of words. The consonants (vyanjanaa) of the Sanskrit alphabet were arranged in the order of the table below.

Katapayadi - Roman letter transliteration for Sanskrit alphabet

• The name ka-ta-pa-ya-aadi  क–ट–प–य–आदि comes from the beginning (aadi) letters in each row. Any number can then be represented by any one of two three or four letters. For example, 1729 can be written as ka-cha-kha-jha ( कछखझ ) or pa-sa-ra-dha ( पसरध ). Any vowel may be combined with each consonant, so pu-si-raa-dhu is the same as pa-sa-ra-dha. This leads to interesting possibilities, where even meaningful words sentences and puns can be formed based on names of numbers.

The Katapayadi system is popularly in use in notation of raga names in Carnatic music, in Venkatamakhi’s mela-karta naming system of the ragas, where first two letters of the raga’s name represent number of the raga. For example, Maayamaalvagowla is the 15^th raga because Maa is 5, ya is 1, so Maaya is 51 (LSD to MSD, here). Dhenuka is ninth (09^th) raga written as Dhe(9)-nu(0). Chaarukesi is the  26^th raga - chaa (6) ru (2).

References

• Facets of Indian Astronomy, by KV Sarma
• Oral Traditions of Sanskrit, Powerpoint of  S Swaminathan
• NPTEL Lectures on Indian Mathematics by Profs MD Srinivas, MS Sriram, K Ramasubramanian

In 1999, I read the Tamil historical novel Sivakaamiyin Sabatham. Set in the seventh century, the novel tells of the siege of Kanchipuram, the capital of the Pallavas, by their rivals, the Chalukyas. It is still one of the most popular books in Tamilnadu, seventy years after its first publication. At one point in the story, Aayanar, an artist and sculptor tells the Pallava king Mahendra Varma, that his deepest desire is to know the secret technology of  Ajanta paintings, which have lasted a thousand years without fading. The paintings of Ajanta are still there, a thousand years after the Pallavas and Chalukyas disappeared.

In Kanchipuram, the Kailasanatha temple built by Mahendra Varma’s great grandson Rajasimha Pallava in the eighth century, has paintings in the Ajanta style, which have sustained damage but what remains hasn’t faded. The contemporary Pandyas built a Jain cave temple of Sittannavasal, which has equally remarkable paintings.

A heritage of science and technology

These struck a severely discordant note. How many advertisements do we see on television for the latest paints that last twenty years? Enamel paints manaufactured in large chemical plants, based on the very latest technology brought to life by the most brilliant chemical engineers of the last century. Their great selling point is that they last twenty years – one hundredth of the two thousand years that Ajanta paintings have lasted, in primitive caves, sculpted by hammer and chisel.

What other remarkable scientific and technological achievements of ancient Indians was I missing?

By sheer coincidence, I happened to attend a series of lectures about the “Oral Traditions of the Sanskrit” language, by Prof Swaminathan, a retired IIT Delhi professor of mechanical engineering. He explained the Siva sutras (also called Maheshvara sutras) and how Panini used them to write extremely compact rules of grammar for Sanskrit. The Siva Sutras and Panini’s sutras reminded me starkly of the Backus-Naur notation, that every computer science or engineering student learns in college. But, wait! What was Panini doing, composing Sanskrit grammar in Backus-Naur notation?

Why is Panini never mentioned in any computer science course? Why is not a single discovery  of Baudhayana, Aryabhata, Brahmagupta or Bhaskara ever taught in a mathematics or engineering course? In any school or college? Seventy years after independence, you can hardly blame British colonialism.

The ignorance is not merely about Indian science, it is about all non European science and technology, in general. Sumeria, Egypt, China, MesoAmerica (Olmecs-Mayans-Aztecs), Persia, all ancient civilizations are totally ignored, and we get an entirely European perspective of all science and technology.

English is the language of science, we are told, though most of the scientific vocabulary is in Greek or Latin. The very names of the sciences Physics, Biology, Zoology, Geology, Astronomy come from Greek. Chemistry, is an exception, adapted from the Arabic word AlChimia (or Alchemy). The  different fields of mathematics, Geometry, Trigonometry, Arithmetic have Greek origins. But Algebra comes from an Arabic word; Calculus from a Latin word. Newton wrote his most famous physics book, “Principa Mathematica de Naturalis” in Latin, not English. When Antoine Lavoisier coined new words for the modern chemistry he discovered, he did not use French; he chose Greek and Latin. English words like soda and pot ash, were Latinized into Sodium and Potassium.

A mathematical vocabulary

Did the Sumerians, Chinese, or ancient Indians use Latin or Greek? Or even need them? Obviously not. It was when I started reading the Aryabhateeyam in its original Sanskrit (with English translation assisting), that I realized what a rich vocabulary we are ignorant of.

Do you recognize the following words: vishkambha, parinaaha, kakshya, vishuvat, karna, jyaa?

How about these words : diameter, circumference, orbit, equator, hypotenuse, sine?

Here’s the stunnner. The first row of Sanskrit words have the exact same meaning as the second row of English words.

समपरिणाहस्यार्ध विष्कम्भार्धहतमेव वृत्तफलम् ।

Transliteration sama pariNaahasya ardha vishkamba ardha hatameva vrtta phalam

Let me explain this Sanskrit statement, word by word:

Sama – equal
Parinaaha – diameter
Ardha – half
Vishkambha – circumference
Hatam – multiply
Eva – exactly
Vrtta – circle
Phalam – result 

Literarlly “Equal diameter-half circumference-half mutliply-exactly circle’s-result”

Rephrased grammatically in English : “A circle’s area equals half the diameter  multiplied by half the circumference”.

This was stated in Sanskrit by none other than Aryabhata. It is the seventh sloka in his Aryabhateeyam.

Let me propose two quick quizzes: there are two lists of names side by side, one European, the other Indian. Just write down what they invented or discovered, as a self-test.

You can the internet to verify your answers. But did you get all answers correctly in the first list? How did you fare with the second list? Did you even recognize all the names? (Confession: I didn’t know three of them ten years ago). If you guessed that Aryabhata invented zero or discovered gravity or the heliocentric theory, give yourself negative marks. He didn’t.

But the people on the second list had one things in common. They all used Sanskrit as the language of science. Why Sanskrit? Sanskrit was not only the language of religion, and literature, it was also the language of several sciences, law, justice, administration, economics, rhetoric, logic, and several arts, namely music, dance, painting, sculpure, architecture etc. It served the same function in India and countries to the east of India, that Latin first in the Roman empire, then in Europe until perhaps the twentieth century; what Mandarin did in China from Confucian times upto perhaps today; what Arabic did in the realms of Islam. It was the link language of a cultural continent, across several kingdoms over the span of several centuries, even millennia.

Consider these somewhat famous books.

I have provided only one example in each field. In reality, each field has several books, written by scholars from various regions or cities, across several centuries. We never hear of them, because over time, Sanskrit has become more alien in India than Greek or Latin.

Now consider that quiz, again. Why is that ignorance of the inventions or discoveries of Europeans considered scientific illiteracy, but ignorance of the inventions of discoveries of Indians considered normal? It may be tempting to Islamic desturction or European colonialism. But I don’t think that is an acceptable excuse, seventy years after Independence.

When most Indians, hear Sanskrit or hear of it, we only hear of it as the language of the Vedas, or at best the language of beautfiul poetry as in Kalidasa or Jayadeva. One popular understanding is that it is a dead language, steeped in the superstition of religion. The only people talking in public about anything Sanskrit are people quoting philosophy; once in a blue moon, perhaps a musician or a dancer. Or, a chorus chanting Sanskrit mantras as background music in a Star Wars movie.

Buddhist and Jain Sanskrit literature

Sanskrit was not the only language in which science was written, in ancient and medieval India. The Jains and Buddhists wrote books on some sciences in several Prakrits, primarily Ardha Magadhi and Pali. They believed that Sanskrit was the language of the elite, and to reach the common man, the local languages should be used. But this soon led to severe fragmentation of literature. The Kushana king Kanishka convened a Buddhist Sangha in Kashmir, at which scholars began to translate several Buddhist canonical texts from Pali to Sanskrit. From then on, several original works, including on mathematics, were composed in Sanskrit also. Similarly, Jains composed Sanskrit works from the fifth century onwards, after the Valabhi Sangham. The first Sanskrit book where mathematics is the primary subject, not a chapter in an astronomy book, is Ganita Sara Sangraha, composed by the 9^th century Jain mathematician Mahavira. A few stanzas of his first chapter, beautifully outline the use and power of mathematics. It should be declared the Mathematics Anthem, and printed on the first page of ever math text book. I suspect Finland or Cambodia will do it, and then India will rush to follow. Here it is, with my translation:

लौकिके वैदिके वापि तथा सामायिकेऽपि य: |
व्यापारस्तत्र सर्वत्र संख्यानमुपयुज्यते || ९

कामतन्त्रेऽर्थतन्त्रे च गान्धर्वे नाटकेऽपि वा|
सूपशास्त्रे तथा वैद्ये वास्तुविद्यादिवस्तुषु || १०

छन्दोऽलङ्कारकाव्येषु तर्कव्याकरणादिषु |
कलागुणेषु सर्वेषु प्रस्तुतं गणितं परम् || ११

Translation

In worldly life, in Vedic learning, in religious practice, 
In business, in everything, Mathematics is useful.

In romance, economics, in music dance and drama,
In cooking, medicine and in architecture, 

In prosody, poetry, logic and grammar,
In all the arts, Mathematics reigns supreme.

The libraires of Alexandria and Nalanda may have been destroyed by iconoclastic invaders, but the library of all Sanskrit knowledge is vandalized every day, by our collective ignorance and negligence.

That is ridiculous. We can change that.

References

• 1.     Facets of Indian Astronomy, KV Sarma, 1975
• 2.     The Aryabhatiya of Aryabhata, Walter Eugene Clark, 1930
• 3.   Mahavira's Ganita Saara Sangraha, Prof Rangacharya, Univ of Madras, 1912 

__________

This was the first of a series of essays published in Swarajya magazine online

Related Links

My blogs on Astronomy and Mathematics

Shilpam Science Sundaram - TEDx lecture at Saveetha Eco Pupil school

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.

Related Posts

Lecture Notes

Blogs on Music

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 18^th and 19^th 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 3600^th 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

Bharata Matha terracotta image

Bharatha Matha Terracota figure of composite unit of India, as visualized by Mahakavi Subramanya Bharathi and his close associates, Mandayam Brothers S Thirumalachari, S Srinivasa Chari and S Parthasarathy Iyengar in the year 1916 during their exile at Pondicherry.  This idol was discreetly smuggled to Madras and taken in procession during Anti British movements. It has lost the Ceylon part which was in the form of a lotus bud at the feet,  during an encounter with British police.

Vande mataram.

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This text above, along with this picture, was shared by VK Srinivasan in our Tamil Heritage Trust Whatsapp group.

The Mandyam brothers owned a Tamil newspaper called Swadesa Mitran, in Madras (Chennai) and they hired Subrahmanya Bharati, the poet, as its deputy editor. Bharati today is primarily known throughout the Tamil speaking world as poet extraordinaire, a Mahakavi, whose patriotic poems about India, the Tamil language, are well known. But he did not earn much from his poetry; his primary income was as a journalist. He is considered THE pioneering journalist of the Tamil language - he wrote political essays advocating the expulsion of British rule (inspired by Bala Gangadhara Tilak), he introduced the first cartoons in Tamil newspapers, he wrote essays on science and technology, translated essays and poems from various languages of India like Bengali, Hindi, Marathi, and English. He was considered an extremist in those days, a follower of Bala Gangadhara Tilak, as opposed to the moderates led by the lawyers of Madras, led by the indomitable V Krishnaswami Aiyar. He introduced the Bengali phrase Vande Mataram into Tamil, without modifying the words, and made it popular. He also published an English newspaper, for years.

A cartoon published by Subramania Bharati Gandhi as a cow, the British as tigers

He formed a patriotic group with VO Chidamaram Pillai, Subramania Siva, Aurobindo Ghosh and VaVeSu Iyer. VO Chidambaram a lawyer of Thoothukkudi (Tuticorin), an old harbour city near Kanyakumari, is famous for starting a shipping company. He was later arrested and sentenced to jail, where he was also sentenced to be yoked (Literally a wooden yoke, like a plough) to an oil-press (instead of a bull) and forced to pull the oil-press.

Bharathi and others condemned this arrest and cruel sentence. The British issued an arrest order for Bharati for one of his essays, and he escaped to Puducheri (Pondicheri) which was under French rule, taking the advise of his friends. He spent nearly ten years in Pondicheri, in self-exile. At some point he was exhausted with his exhile and came back to British India - that is Madras province. He was arrested in Kadaloor (Cuddalore) and sentenced to jail, but was released after promising not to write against British government policies. Sadly he died a few years later. 

The Mandayam family, which ran the SwadesaMitran newspaper, and had hired Bharati as a writer and assistant editor, somehow inherited the terracotta idol of Bharata Matha. Decades later, after independence, N Balasubramanian, a mathematician and cryptologist, and an avid lover of Tamil literature and especially Subramaniya Bharati, was an active membre of Delhi Tamil Sangam. By the 1960s, a significant number of Tamils had moved to Delhi, a vast number of them employed as bureaucrats in the Government of India, and quite a few in academic circles also. The Delhi Tamil Sangam, regularly held meetings in which they discussed literature, hosted visiting writers, speakers, artists, etc; conducted music concerts (mostly Carnatic music) and Tamil plays and so on. Balasubramanian, who wrote under the Tamil pseudonym Nagupoliyan was one of the very active members of this group. On one of his travels, a member of the Mandayam family gave him the Bharatha matha statue for safekeeping. From then on, every monthly meeting of the Delhi Tamil Sangam started with a prayer and puja to this Bharata Matha statue. In 1982 the centenary year of Subramaniya Bharathi, year long celebration called Bharati 200 was conducted. It was called Bharathi 200 rather than 100, in the hope that Bharathi would be remembered for at least another hundred years. 

When Balasubramanian retired and settled down in Madras (Chennai), he brought back this image with him and somewhere along the way, the Bharata Matha statue returned to the Mandyam family. During the condolence meeting of R A Padmanabhan in 2014, who wrote a biography of Subramaniya Bharati, I met one of the stalwarts of the Mandayam family - Mandayam Parthasarathy Iyengar, then around 96 years old; he was one of the speakers.

Mandayam Parthasarathi Iyengar at  condolence meeting of RA Padmanabhan, 2014

I met "Nagupoliyan" Balasubramanian around 2010. We called him Balu sir and learned about the various aspects of his life only later. He announced a weekly Sanskrit class then, and I began to attend it regularly until 2015 or so. I didn't learn much Sanskrit, but learnt a lot about sanskrit and also learnt a lot about Bharati and a many fascinating aspects of Tamil literature from him. At this point he only had a photo of this Bharata Matha statue with him. In December 2010, he decided to donate a collection of his books on Bharati and his Bharati memorabilia, including clippings from several newspapers and magazines to the Mahakavi Bharati School in Kasuva village, near Thiruninravur, run by Sevalaya. I accompanied Balu sir and several of his writer friends, where stalwarts like writers Ja Ra Sundaresan, and Rani Maindan spoke. Balu sir had already donated some of his collection to the Bharati Illam, a memorial house in Tiruvallikeni near the Parthasarathi temple, and they are on display there.

படங்கள்: இரா.விஸ்வநாதன்

ராஜசிம்ம பல்லவேச்சுரம் - கும்பாபிஷேகம் : படம் - இரா. விஸ்வநாதன்

2017 ஜூன் ஐந்தாம் நாள் காலை, காஞ்சிபுரம் சென்று ராஜசிம்மபல்லவேஸ்வரத்து கும்பாபிஷேகம் காணலாம் என்ற எண்ணம். மதியம் 12 மணிக்கு என்று தெரிந்து, டிவியில் அமெரிக்க கூடைப்பந்து போட்டியை பார்த்துவிட்டு 9 மணிக்கு கோடம்பாகத்தில் பஸ் ஏறினேன். போரூர் சென்று அங்கிருந்து ஏசி (குளிர்சாதன) பஸ்ஸில் போக திட்டம். (இந்த வியாசம் முதிலில் முகநூலில் எழுதியப் பதிவு. இன்று ஜூன் 7 2024ல் இங்கே வலைப்பூவில் பிரதி.).

போரூரில் ஒரு மூடிய கடையின்  வாசற்பந்தலின் நிழலின் அருமை வெயலில் தெரிந்தது. காத்து காத்து நின்றாது தான் மிச்சம். பூந்தமல்லிக்கு செல்லும் பல பஸ்களும் திருப்பெரும்புதூருக்கும் திருவள்ளூரூக்கும் செல்லும் சில பஸ்களும் வந்தனவே தவிற, தி நகரில் புரப்பட்டு காஞ்சிக்கு செல்லும் ஏசி பஸ் ஒன்றும் வரவில்லை. அசரீரி விசையான கைப்பேசியில் நண்பர் விசுவநாதன் காஞ்சியை தான் கிட்டத்தட்ட சேர்ந்துவிட்டதாக சொன்னார் (அவர் பகிர்ந்த படங்கள் மேலே). நான் காரில் வருகிறேனா என்றும் கேட்டார். போரூர் அடைந்த போது 9.30 இருக்கும். 10.15 ஆனதும் கொஞ்சம் பொறுமை இழந்தேன். நடு நடுவே நண்பர்கள் பலரும் கைப்பேசியில் அழைத்து பிறந்தநாள் வாழ்த்து சொன்ன அன்பு தென்றலிலும் வாட்சாப்பிலும் மெசஞ்சரிலும் வந்த வாழ்த்து மழையிலும் நனைந்தேன்.

ஸ்ரீபெரும்புதூர் பஸ் ஒன்று வரவே அங்கிருந்து காஞ்சிக்கு பஸ் பிடித்து போகலாம் என்றெண்ணி ஏறினேன். கண்டக்டரிடம் காஞ்சிபுரம் ஏசி பஸ் சேவை நிறுத்தப்பட்டதா என கேட்க, போரூர்ல் நிரைய பஸ் வராது, பூந்தமல்லியிலிருந்து நிரைய பஸ் இருக்கு என்று அவர் பொத்தாம் பொதுவாக சொல்ல, எதிரில் காஞ்சியிலுருந்து தியாகராய நகர் செல்லும் ஏசி பஸ் தெரிந்தது. ஆஹா, பின்னாடியே ஏசி பஸ் வந்து நாம் ஏறிய பஸ்ஸை தாண்டிவிடும் என்று நினைத்து முற்போக்கு சித்தம் துறந்து பின் ஜன்னல் வழியே ஆந்தைப்போல் கண்விரித்து பின் தொடரும் நிழலின் குரலை, மன்னிக்க, பின் தொடரும் பஸ்ஸின் பலகை வட்டெழுத்தை வாசித்தே பயணித்தேன். பூந்தமல்லியில் ஆமை வேக நெருக்கடி. திருக்கச்சிநம்பியின் நமட்டு சிரிப்பு கேட்டது போல் ஒரு லேசான பிரமை. 11 மணி ஆகிவிட்டது. அடுத்த பத்து நிமிடத்தில் ஏசி பஸ்ஸும் வரவில்லை, பூந்தமல்லியிலிருந்தும் காஞ்சிக்கு எந்த பஸ்ஸும் புரப்படவில்லை. இனிமேல் சென்றால் 12 மணி கும்பாபிஷேகத்தை நிச்சயம் காணமுடியாது என்று வருந்தினேன்.

அசரீரியாக வேறு ஒரு குரல் கேட்டது போலிருந்தது. செல்போனில் அல்ல,

உள்மனதில்.

ஆதி முனிவர் கன்வரோ, ஆதி மன்னர் துஷ்யந்தரோ அசரீரி குரல் கேட்டால் ஆச்சரயமில்லை. அது புண்ணியம் நிரைந்த க்ருத யுகம். யாரிந்த குரல் என்றால் பல்லவேசுவரன் கைலாசநாதனே பேசினான்.

அவன் கூறியதாவது: “சிறுபிள்ளாய் (“youth” என்று பொருள்), இன்று நீ கச்சிப்பேட்டு பெரிய திருக்கற்றளியில் நடக்கவிருக்கும் கும்பாபிஷேகத்தை காண நினைக்கிறாய். ஆனால் என்னை என்றும் எந்தை எந்தை என்றழைக்கும் ஒரு பித்தன் அவ்வூரிலேயே இருந்தும் இந்த கும்பாபிஷேகத்தை காண இயலாமல், தன் சொல்லால் பக்தியால் சித்ததால் பாடலால் நடனத்தால் பணிவால் பணியால் பண்பால் அன்பால் அறத்தால் அறிவால் ஆற்றலால் தெய்வத்தமிழால் வேதமொழியால் விளக்கால் விளக்கத்தால் வழக்கால் வழுக்காமல் வழிப்பட்டு வருகிறான். அவன் மனக்கோவிலில் தான் எனக்கின்று மகத்தான கும்பாபிஷேகம். அதைக்காணும் அகக்கண் உனக்கில்லை. அதனால் அத்யந்தகாமன் எழுப்பிய கற்றளிக்கு நடக்கும் கும்பாபிஷேகத்தையும் நீ காணவேண்டாம்.” 

எனக்கு எப்படி இருக்கும்?

“கைலாசநாதா! காஞ்சி மகாமணி! தர்மநித்யனே! சங்கரநாராயணுனுக்கு தொழில் செய்யவேண்டியதால் அவரால் உன் கும்பாபிஷேகம் பார்க்க முடியவில்லை? எனக்கோ எந்த வேலையும் இல்லை. என்னை தடுப்பது என்ன தர்மமா?”, என்றேன்.

“நாற்பத்தி எட்டு வயதான உன்னை சிற்பிள்ளாய் என்றேன். அது மட்டும் தர்ம்மா?” என்றார் ஈசன்.

“லலிதவிலாசா! எனக்கு கேள்வி கேட்கத்தான் தெரியும்,” என்று முனக, பல்லவ சிற்ப்ம் போல் சிரித்தான் பல்லவாதித்ய பரமேசன்.

“காலக்கோபா! காரணகோபா! என் மேல் என்ன கோபம்?” என்று தொடர...

“கால தாமதா கோபா! உன் மேல் எனக்கேன் கோபம்?”

“நான் காஞ்சி போகிறேன் காஞ்சி போகிறேன் என்று ஊரெல்லாம் சொல்லிவிட்டேன், அது பொய்யாகிவிடுமா? இஷ்டவர்ஷனே! ரிஷபலாஞ்சனா! அதை நீ அனுமதிக்கலாமா?”

“அது பொய்யாகவில்லை. நீ வெயலில் நன்றாக காஞ்சி போய்விட்டாய்” என்றான் அர்த்தபதி. “எனக்கு இவ்வுலகில் ஈடில்லாத அந்த பக்தன் சங்கரநாராயணன் என்ன செய்கிறான் தெரியுமா? இன்று என்னைப் பாடாமல் உன்னை பாடுகின்றான்,” என்றார்

இது எப்படி நடக்கும் என்று ஒரு சின்னசம்சயம் தோன்றிற்று. சின்னசம்சயன் மறைந்தான்.

படம்: விஜய் பட்

அசரீரி விசை மணி ஒலித்தது. சங்கரநாராயணன் இயற்றி முகநூலில் பகிர்ந்த கவிதையை நண்பர் விகே ஸ்ரீநிவாசன் வாட்ஸாப்பில் அனுப்பியிருந்தார்.

இதோ அந்த தடுத்தாட்கொண்ட கவிதை.

கோபுவுக்கு ஒரு வாழ்த்து..

 

நாவின் கிழத்தி நற்பயனோங்க

பாவின் முறைமை பணியொடு சிறக்க

ஓங்கு கலைகளும் ஒருங்க வாய்த்து

தாங்கும் தடவென தாரணி தன்னில்

அரங்கரத்தினத்துக் கருமகனாகி

சுரங்கமெனவே சூழ்கலைக்கிடமாய்

பல்லவர் கலையும் பாவலர் நிலையும்

எல்லையில் வானியல் இயலும் சிறக்க

தமிழ்பாரம்பரியத் தகவுடை நிலையின்

அமிழ்தாய்த் திகழ்ந்து ஆய்வுகள் சிறக்க

இராசசிம்மனெனும் ஈடில் அரசன்

பராவிய எழுத்தின் பாங்கறி நிலையும்

வராகமிகிரர் வரைந்தளித்திட்ட

விராவும் வானியல் விழைந்த திறமும்

கலைபல மலிய கவித்திறம் பொலிய

அலையென புகழும் அடர்ந்த விதமும்

எந்தனைப் போலே இத்தரை மக்கள்

விந்தை யடைந்து விழித்திரை விரிக்க

சிந்தும் புன்னகை சீரெனத் திகழ

முந்தும் புகழோ முகிலையும் கிழிக்க

என்றும் இளமை எழிலொடு திகழும்

மன்றம் பலகாண் மாண்புடை கோபா

நன்றென நாட்கள் நலம்பல பயக்க

என்றும் சிறப்பொடு இனிதென வாழி....

 

Why do we admire the correction of a wrong more than a hundred implementations of the right? We seem to expect the latter so much that we take it for granted.

Why is that most of our fiction, stories and movies celebrate violence in the guise of punishing the wrong doer far far more than very non-violent acts worthy of appreciation? Is it because fiction exaggerates the rare which is what we like to see, rather than commemorate the good which is somehow boring by frequency.

We see the same in art. A hundred beautiful swans is not worthy of appreciation - they are merely decorative, but one swan among crows or pigeons or ducks is considered worthy?

People go on forest tours to see lions and tigers and rhinoceros and deadly animals in the wild but take no cognizance of cows and sheep and hens and cats and dogs etc which are part of  our daily lives. We enjoy flower shows of rare flowers much more than the dozens of beautiful flowers we use and see all around us daily. In an era of plentiful cheap but still excellent music, vast numbers of us would rather pay a huge amount to see a concert in person. Same with sports. The primary reason is excellence and rarity, obviously; but why do excellent things lose their specialty simply by being common?

I can understand rarity is why we appreciate beauty, why do we appreciate cruelty or ferocity or brutality - simply because it is rare?

This seems to pervade human experience across cultures and across time and across a wide variety of philosophical perception and influence. 

Economics have noted the phenomenon called "loss aversion." The vast majority of people are far more offended when they lose something of even small value than they are delighted when they gain something of much higher value. For example, people feel worse about losing ten rupees than they feel happy unexpectedly getting something worth hundred rupees, even unexpectedly.

What I have described before this seems to be a variation of this "loss aversion" concept, except in other fields. In almost every field of life.

 

TiruppudaiMarudur Paintings by Prof Baluswamy

On May 1, 2016 I attended the launch of a Tamil book titled “Chitrakoodam – Tiruppudai Marudur paintings.” The book was authored by Prof Baluswamy of Madras Chrisitian College, Tambaram, whose earlier books “Arjunan Tapasu” and “Krishna Mandapam” also about famous monuments in Mamallapuram are outstanding works of research. Speakers included botany professor Dayanandan of MCC, artists Trotsky Marudu, Smt of Hindu Religious and Charitable Endowments department, photographer Narasimhan and conservator and documentary film-maker MV Baskar. These are notes from their talks which I posted on Facebook, on that day, which I have now copied and edited for this blog. Prof Baluswamy himself delivered a vote of thanks, which I don’t seem to have taken notes for. The event was the Goethe Insitute auditorium, Rutland Gate, Nungambakkam, Madras.

I first heard of these paintings at the Tiruppudaimarudur temple from Prof Baluswamy himself at the 2012 Tamil Heritage Trust Pechu Kacheri on Paintings of India, which was held in the Tatvaaloka auditoirum, Eldams Road, Teynampet in Chennai. These paintings and the unusual military history they recorded were part of his talk which was about Nayak era paintings of the Tanjavur and Madurai Nayaks. It was recorded along with other talks at the Pechu Kacheri but none of these have been yet edited and made available to the public.

The book was only released last year and was available at the Chennai Book Fair in Nandanam. An earlier book on these paintings by Dr Kannan director of Chennai Egmore museum, was presented to me as a gift around this time, for my talk and guided tour about the Amaravati gallery of the Egmore museum, by the then Director, Dr Kavita Ramu. Dr Kannan who reshaped the Amaravati gallery also undertook a restoration/renovation effort of these murals and has recorded them in his book.

The notes from the talks at the 2016 book launch follow:

Prof Dayanandan, MCC

Lawrence Surendra played a major role in developing this book, "Chitrakoodam - Tiruppudaimarudur Oviyangal", by using MCC as a powerful resource to research and collate.

Baluswamy is our treasure. That others share him invokes jealousy and pride. We had a great asan at MCC in 60s Gift Siromoney, who pulled all of us into several fields. we thought he had reached a peak with Arjuna's penance, but he has breached new barriers with this book on Tiruppudaimarudur paintings. He has a wide variety of knowledge in depth.

Nagaswamy threw a bomb that Rajasimha built everything in Mallai. Gift Siromoney measured all sculptures with scales. We discovered that no male was depicted before 720. Baluswamy went beyond all this. He brought whole new theories to scholarship on Mallai. Using biology, sculpture sense etc. He discovered that the painting in Tiruppudaimarudur depicted the Tamraparni battle between Vijayanagar and Travancore. After the advent of Portuguese. I discussed Columbian exchange with Baluswamy, after which pineapple potato chili etc entered India.

Sangam Era marudu tree would have been lagerstomia. Sthala trees are not more than 200 or 300 years old, opined a botany professor in Madras University.

Baluswamy makes the distinction between mythology and history when teaching. No humans in India before sixty thousand years. So please forget Lemuria etc. When Africans settled the world. The common mother of all living humans lived in Africa 180000 years ago. Based on mitochondrial evidence.

Who ordered the paintings at Tiruppudaimarudur? Did the painters paint what they were ordered to? The book will discuss all these.

Panelists and audience - photo by VK Srinivasan

Trotsky Marudu, artist

I grew up with the Nayak paintings around poRRamarai tank in Madurai Meenakshi temple. All now eradicated. This tragedy bothered for a longtime. Western realism combined with Indian artistic tradition only in the Nayak period. My predecessors like Adimoolam worked at Weavers Institute. Government photographers captured paintings of Alagar Koil, Avudaiyar Koil etc and they used those designs in weaving textiles.

Even Adimoolam has not seen the Tiruppudaimarudur paintings. MV Baskar, Sarangan, myself and Baluswamy had the opportunity to see them after 2000. In 1990s I could not go past the first level and it was too dark to see these paintings. After the Danielles, Europe saw India in pictures. Parsi theater gave another visual perspective to Europeans. Later Ravi Varma printed his own paintings, made with costumes used by Parsi theater, most of which became popular across households in India. Marathi cinema 's look influences films of South India films now, having few Tamil identity. Dadasaheb Phalke Ravi Varma's Contemporary, and the artists who worked in Bombay film studios determined the look of Telugu and Tamil cinema. Film arts, calendar art and magazine art became the defining art of Tamil country (disconnected from the temple art of the Nayak era).

For several decades maps and images have been eradicated from Tamil books as cost cutting measure.

European books for children had paintings from Egyptian temples and pyramids and much less text. Fourteenth century venetian art decorated such books about Egypt and even Hollywood actors. Indian books do such a lousy job. Hyder Ali for example depicts him practically as a beggar though there are wonderful pictures of him. Sadly Tamil culture is mostly verbal. We pay poor attention to visual art.

Thadagam published a very visual book and they are doing this with Baluswamy's book on Tiruppudaimarudur. This is a treasure for the world at large not just Tamils.

Smt Kavitha of HR&CE

Of the thirty eight thousand temples under HR&CE about fifty temples have paintings like Tiruppudaimarudur. We have permitted several people to photograph paintings for scholarly study. Our department is blind to a lot of technology, and have inadequate in situ scholarly talent. We are now restoring temple paintings. INTACH restored kuRRalam Chitra Sabha, then another group complained that INTACH had ruined them by using very bright colors.

There are wooden panels at every level in Tiruppudaimarudur gopuram. Insect infestation and bats were big problems. Water seepage is another problem. We used to use cement, then combination mortar, and now we use lime mortar. We don't touch up the paintings. In ASI chemists do the job. Our sthapathi said Silpa Sastra shows several sources of paint including sand and stone colors, not just vegetable dye.

There are PhDs who have published with no reference to the temple or HR&CE. Why can't you give copies to the temple and our department??

Please cooperate with us. Don't have an antagonistic approach. We find it very difficult to find talent for restoration. A text called "Aalaya nirmana bimba Lakshana" shows us how to restore temples. Materials die, just preserve the monument, not just the materials. Expertise is hard to come by. Maintaining thousands of temples have to be maintain 38,000 temples with income from 2000 temples. It's not our intention to destroy, we do it out of ignorance.

Or by good intentions to save a monument or protect public from collapsing roof etc (one such attempt at Srirangam thayar shrine caused the paintings to crumble).

I hope at least Prof Baluswamy will acknowledge HR&CE

Episodes from the live of Gnanasambandar - a slide from a PPT

A painting of Portuguese horse trade

Photographer Narasimhan -  Documenting the paintings

Our long time dream is getting fulfilled. I must acknowledge one who has not been acknowledged by other speakers. Mrs Baluswamy approved all the great expenses involved. Two types of people visit Baluswamy, those who go to learn and those who go to plagiarize.

Amutharasan came forward to publish the book. This pre publication is a special effort.

The technology to document was brought to us by Sarangan and Baskar.

Tiruppudaimarudur is a rare temple to still have original Nayak Era paintings. Most temple paintings have suffered great change. We have documented Alagar Koil paintings. They are valuable historical evidence. Baluswamy's effort to understand the paintings has been phenomenal. Investigating the Tamraparni war took four years. Lots of discussions with Sanskrit scholars to understand historical narrations of the war. Retired Supreme Court Justice Ratnavel Pandyan has been wonderfully helpful in this research.

We held several conferences to convey the importance to students and public. We ran a workshop at Srirangam for temple executive officers to our preserve sculptures and paintings.

One of the quite but wonderful servants of the documentation of this book is Mr Uthraadam, a PhD student of Baluswamy. This books is a milestone in tamil history.

MV Baskar - Visual documentation

Thirteen years of Hard labor. Documentation is not mere photography but Archaeometry. Digital cameras of 8Megapixels entered India in 2003. Now we shoot with 80Mp cameras. We composed each photo of A4 size, so a wall painting will be photographed as sixty different shots. Lighting is uneven so the same red may show up in different shades in adjacent shots. We used gray cards to give us color context.

Most people shoot the most photogenic scenes. We shot everything even the missing pieces which had only damaged wall or even graffiti, so we can reconstruct the painting. All paintings tend to be narrative paintings in temples, not pattern paintings. Vasanta mandapam of Alagar Koil has lovely Ramayana depiction. Tirugokarnam has Ramayana as we exit the temple, a structural narrative.

We had two photographers just for paintings and two for all other aspects. Digital tempts them to shoot limitlessly. We threw away two days worth of photos because we could not put them together.

So we had to shoot after planning, by mapping the pictures first. Cameras have no file structure. We had to map and key in advance and then collate photos in that order.

This is scalable vector art, machine readable. I worked with a Kalamkari artist from Andhra. We need pigment analysis and such equipment is only in Aurangabad and Bombay. We picked up all the fallen paint flecks and put them in zip lock bags.

Every painting in every temple has been over painted. They can only be seen with infrared photography. An infrared camera is twenty thousand dollars. Now it is much cheaper. We can rig one. I've shot Ladakh Buddhist monastery paintings with infrared cameras and they are also multi layered over time.

Academically, art and design schools lack tangible programs to educate students. We need multiple accredited programs at respected colleges to do this. Paintings also are accompanied by inscriptions or text.

The paintings depict several musical instruments. We recorded sounds from such musical instruments to create a multi media presentation. History can be far more engaging than only Archaeology. Money shortage is no longer an issue.

I'm delighted Art Painting has finally come to Tamilnadu. This is quite popular in Bombay Delhi etc but so far unsuccessful here. Hotels ask for paintings but haven't paid and ran some people bankrupt. But that is changing

----End of Lecture notes---

Acknowledgement I downloaded some photos of Powerpoint presentations on screen at the lecture, taken by VK Srinivasan during the book release and posted in the comments of my Facebook and added them in this blog. I presume the original photos used in the Powerpoint are by photographer Narasimhan, who was one of the speakers.

Related Links

Prof Baluswamy on Arjuna’s Penance in Mamallapuram

Prof Dayanandan’s lecture on Evolution at Varahamihira Science Forum

MV Baskar on Preserving Ramayana Murals at Tamil Heritage Trust

Index of My essays on Art

Index of  Lecture Notes in this blog

I attended the India Science Festival held at IISER, Pune January 20 and 21, 2024. One of the lectures was by Prof Alan Harvey, University of Western Australia, titled “Why does music exist?”  These are my notes from that lecture.

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Alan Harvey's lecture at ISF pandal, Pune

Why does Music exist? Speaker - Alan Harvey

Music is a science,not an art. Maybe we shouldn't think of art and science as different things

How do I remember so many songs but don't remember much prose ?

Theres something special about music, which fires the communication stream, and oxytocin is major factor. Oxytocin is involved in the plasticity of memory, which works with music in a way language alone doesn't.

The human species is the only one that sing in harmony and move synchronised like in dances.

Music fires different regions of the brain as can be seen in MRI scans that scrambled noise fires.

There may be an evolutionary reason why the ability for music evolved, and we have several conjectures, but nothing has been proved.

A slide from Alan Harvey's lecture

You shouldn't use the word music unless you refer to modern music which distinguishes compositions with song and instrument distinct from language. To call birdsong or tweets of insects or other creatures as music is absurd.

Music is different from just poetry, even poetry with rhythm, even poetry with pitch changes.

But scientists have not studied or recorded brain patterns based on poetry, as much as with music, and it's worth exploring.

Other Lecture Notes

Other essays on Music

 

Edison and his phonograph

In the 1870s three revolutionary technologies shook the world and changed it forever :

1876 : telephone

1877 : phonograph

1879: electric light bulb

All were basically leap-frog innovations, but based on existing science and technology.

The contrasting impact of the three were fascinating.

Phonographs which earned Edison the moniker Wizard of Menlo Park, completely stalled, because amplifiers were bad, duplication was not possible, etc. That problem was solved 20 years later by invention of the triode (ironically inspired by Edison’s electric bulb).

Telephones revolutioninzed business, government, industry, banking and military, but took nearly forty years to become consumer goods.

The electric light was utterly transformational across every facet of society. Until Edison’s light bulb, electricity was also mainly industrial or business oriented. Heavy magnets at harbours and factories, and the telegraph were the only public benefits of electricity until then.

Edisons light bulb changed all this. It brought electricity into every urban home and office within 20 years in most of the West.

It became a platform for a whole horde of technologies as transformational as the steam engine earlier, and transparent glass even earlier.

Electricity’s impact was rivalled only by the arrival of petrol/diesel engine cars, also invented in the 1870s but which had no or very low impact for nearly 25 years.

Electric cars, trains, buses all came into use, around this era, but were of poor quality and not very widespread. They were all overtaken by petrol based vehicles by 1920 or so and electric vehicles did not challenge petrol/diesel vehicles until Elon Musk’s Tesla transformed the industry in the 2010s. Henry Ford actually worked in Edison’s electric companies, rising quickly to Chief Engineer, before quitting to revolutionise petrol based cars and make them widespread.

Edison's telephone: Picture from Internet

The triode, born out of the diode, invented by Ambrose Fleming (more famous for Fleming’s left hand rule), was discovered to be an amplifier of current, which became very useful for radios, invented shortly afterwards. Then the triode amplifier became useful to amplify the sound in phonographs. Edison’s original tin drum phonograph did not take off, but Zenophile Gramme invented the flat disk version, now popularly called gramophones. By 1910 mass manufacturing of gramophones combined with amplification to make the sound output of gramophone, made them explode in popuarity – nearly two decades after Edison and Gramme’s inventions. I suspect Edison himself had switched over to focusing his efforts on the light bulb – and after it the development of the entire electrical system, primarily because the phonograph evoked more wonder than sales.

The telephone itself, was also improved by Edison, especially by addition of  a battery, providing a stable current rather than using a windup mechanism to power the transmission of electricity across telephone lines. Alexander Graham Bell’s company AT&T, bought Edison’s phones to sell to their customers – became more the telephone exchange rather than equipment manufacturer.

Ironically phonographs/gramophones succeded in the one field Edison did not imagine at all – recorded music. Again ironically, Edison worked on the cinematograph and movie making, but again failed to understand what entertainment would be popular. The movie making world ran away from New York, where Edison was based, to California – too far even by train – to escape Edison’s attempts to monopolize the field.

It is fascinating how the three inventions panned out. Rarely is their comparative history narrated together.

But the phonograph, lightbulb, electrification, (and also triode, radio, airplane etc.) were inventions that did not come from the learned halls of brilliant academics with doctorates in philosophy or from the wise poets and artists who bore on their shoulders the learning of a hundred previous generations, but basically the most uneducated/ self-educated, hyper motivated, relentless, incredibly ingenious people like Edison, Ford, Benz, Daimler, Westinghouse, Lee de forest, Fessenden, etc.

Thorstein Veblen, the economist, predicted in 1899 that engineers would run the world , because they were the only people who understood how everything worked.

He was wrong, IMHO. Most engineers don't understand people or money.

Inventors and Discoverers

I attended a lecture by Prof Wayan Dibia, a scholar and choreographer from Bali, Indonesia, at the CP Ramaswamy Iyer art gallery on Saturday January 6, 2024. He gave a very brief introduction on Balines temples, their layout and dances at the temples. Dance is a very important part of the ritual worship in Bali it seems. Bali is an island in Indonesia with nearly 95% Hindu population. Indonesia is the country with the largest Muslim population in the world, but is fairly cosmopolitan and celebrates its Hindu past.

These are notes from Prof Dibia’s lecture.

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Balinese Hindus greet each other with the phrase “Svasti Astu”,  a Sanskrit phrase which means “May you be well”. Rather than namaste or namaskaram, the common Hindu greeting in India.

A typical Bali temple

Bali is refered to as Island of the gods. Every house has a family shrine. People pray at these family shrines every day and visit the temples on important occasions.

There are four types of temples in Bali:

1. khayangan jagat,. Worhsipped by all Hindus.

2. Khayangan Desha , three main temples in each village(desha) worhsipped by villagers

3. Pura Swagina, worhsipped by people of same profession

4. Pura dadia or Pura kawitan worhsipped my members of same clan

Each temple has three shrines - one each for Brahma, Vishnu, Shiva. Only if all three temples are present can a settlement be called a village. Brahma and Vishnu temples are in middle of villages. Siva temples near graveyard.

Temples usually have three sections, uttama mandala, madhya mandala and jaba mandala. Basically there are three courtyards in the temple.

Odalan is a Bali tradition; it involves rituals requiring sacrifices and offerings of many types at temples. The entire group of villages wearing tradition attire, goes in procession to a temple carrying fruits, flowers and other offerings. Musicians and dancers usually follow at the tail end of the procession.

A 1971 seminar was conducted by Bali Provincial government to categorise dances as sacred vs tourist entertainment.

The seminar declared that the three major types of Balinese dances are:

·        Wali dance, which is performed in main courtyard of a temple.

·        Babali dance, performed in second courtyard

·        Balih balihan dance, which may only be performed outside the temple.

There are Rejang dances performed by women and baris dances performed by men

Babali dances usually tell a story and are dramatic. They have selected dancers. The story is usually Ramayana or Mahabharata.

There is custom called Sidhakarya with a history. A priest from Kalinga (Odisha state in India) came to Bali, in 16th century. The king ignored him, but then several aspects went wrong with ceremony. The king then recalled the priest, regretting his rudeness. Since then a priest is always involved in ceremony called  Topeng Sidhakarya.

Some temple dances are declining. Some are extinct or very commercialized. Most temples don't have vigrahas of the murthys because they are kept in the house of the priests. Only during odalan ceremony is the vigraha brought out and kept in the temple.

Icons in temples

The above photos were snapped by me from the slides on the screen during the lecture.

Audience Q&A

Smt Nandita Krishna, who runs the CPR center, said she was not allowed into a Bali temple with salwar kameez or saree, the two popular costumes among Hindus in India. They expected her to wear the Balinese costume which she said she didn't have. Ravishankar Thiyagarajan, of Tamil Heritage Trust, said they were given some sarong and costume to enter the temple on their recent Bali visit.

Mohan Krishnamurthy of Gandhi Center, T Nagar, and THT, asked whether Bali temples had agamas. Nandita Krishna replied that they basically follow Indian agamas and traditions only, but Prof Dibia didn’t comment.

Sri Mohan also asked about rice given to devotees at temples. Prof Dibia said rices is sacred was offered as prasad. 

Their troupe will perform a Balinese dance at the Music Academy, Madras, on January 15, 2024, on the this very theme Arisi (Rice)

Sri Sowndarrajan, also of THT, asked why some gates have just two pillars. This is just an architectural affectation, was the reply.

To a question about what kind of music and songs were played, a friend and colleague of the speaker said that they start music with nattai ragam and end with surutti ragam. But they mainly play instrumental music. Bali unlike India, Bali didn't have bhakti movement, which produced a lot of poetry set to music. So in Balinese dances, there was no sahitya (words and poetry) for the music, only melody. 

Other Lecture Notes

Essays on Art

I visited the Swami Vivekananda Rural Community college near Pondichery with prof Swaminathan and his wife on Monday September 25th.

This is being run as a technical education center for lower middle class students who struggle to continue education for financial or other reasons. The college offers practical training in various mechanical, electrical and computer engineering to their 150 students. Their labs are now sponsored and equipped by companies like Dell Siemens Renault Yamaha etc. 

Students learn such skills as 

• repairing or servicing motorbikes, scooters, cars including 
• repairing and servicing plumbing equipment
• software skills such as Microsoft Office, Tally
• repairing and servicing desktop and laptop computers and their electronics
• design of pamphlets posters invitations programs etc
• medical assistance including for elderly people, pregnant women

Several of these students go on to become entrepreneurs, while quite a few find employment in these companies or others around Pondichery. Several girls are also enrolled, and there is an entire room of both mechanical and electrical sewing machines where they learn various tailoring and embroidery techniques.

We were given a thirty minute tour of several of the labs and classrooms, by a member of the staff Mr Santosh. His pride in the service he was doing and the success of some of the college's students was palpable in his voice. Since we arrived late in the day most of the students had left for the day, and we only spoke very briefly to one student.

The college was founded by Mr Subramaniam, a former colleague of Prof Swaminathan at Delhi, who literally asked for five and ten rupees donations to start this college several years ago. Mr Subramaniam's whole family including wife, son and daughter in law are engaged full time in running this college. They say they dont seek or take any assistance from government organizations. Mr Santosh mentioned that besides teaching the skills they also give basic etiquette training, as in dressing properly, showing up with a presentable appearance, proper behaviour and speech etc.

The college reminded me somewhat of Sevalaya's Mahakavi Bharathi school near Thiruninravur, run primarily by the efforts of Mr Muralidharan.

I must confess, that while the efforts of the founders of such charitable organizations dedicated to education are extremely laudable, I wonder if they are not taking on too big a challenge, trying to outdo what state or local governments are providing with a much larger financial base. I wonder about the sustainability of these ventures past the energetic lives of their founders or most active organizers. Then it occurs to me even governments and private companies have exactly the same issue - in fact, thanks to elections, governments have even briefer tenures : it is the bureaucracy that carries on the larger and somewhat onerous task. Regardless of my prejudice or skepticism they seem to be doing an amazing job. 

School Education in Tamilnadu - My translation of Jeyamohan's essay 

Education - a Sanskrit proverb

தியாகராய நகர் தக்கர் பாபா பள்ளிக்கூடத்தில் தமிழ் பாரம்பரிய அறக்கட்டளை அரங்கில் இந்திரா பார்த்தசாரதிசிலப்பதிகாரத்தை பற்றி பேசிய நாள். கடைசியில் அவர் மேடை இறங்கி ரசிகர்களை சந்தித்தப்போது, ஒருவர் “என்னை தெரிகிறதா?” என்று கேட்டார். பார்த்தசாரதி தயங்க, அவர் “சார்வாகன்” என்று லேசாக சிரித்துக்கொண்டே தன் பெயரை தெரிவித்தார். அது அவரது இயற்பெயர் அல்ல, சிறுகதைகள் எழுத சூட்டிக்கொண்ட புனைப்பெயர். தன்னோடு வந்திருந்த தன் சகதோரரை அறிமுகம் செய்தார்.

சிலப்பதிகார சிக்கல்கள்- இந்திரா பார்த்தசாரதி உரை சார்வாகன் முன்வரிசை, மோகன் ஹரிஹரன் நடு வரிசை மியூசிக் கண்ணன் பின் வரிசை

தம்பி கொஞ்சம் நடிகர் நாசர் சாயல். அப்பொழுது அவர் பெயர் நினைவில்லை; ஆனால அந்த ஆண்டு இறுதியில் த.ப.ஆவின் கலையுலாவில் கலந்து கொள்ள ஆர்வம் தெரிவித்து, பேரா சாமிநாதனிடம் சில தகவல்களை கேட்டுக்கொண்டார். அக்டோபாரில் கலையுலா உரைகள் தொடங்கியபோது அமெரிக்காவில் இருந்து ஈமெயில் அனுப்பி, தன் உதவியாளரை (அவர் பெயரும் சாமிநாதன்) அனுப்பி வைத்தார். வாராவாரம் பினாமியாக பாவம் அந்த சாமிநாதன் எங்கள் கூட்டங்களில் கலந்துகொண்டார்.

ஓரிரண்டு மாதம் கழித்து நேரிலேயே வந்தார் சார்வாகனின் தம்பி. தன் பெயர் ஹரிஹரன் என்றார். மெதுவாக, ஹரிஹரன் என்பது தனது தந்தையின் பெயர், ஆனால் தானும் தன் மூன்று சகோதரர்களும் தங்கள் இயற்பெயரை சொல்லாமல் தந்தை பெயரான ஹரிஹரன் என்றே அறிமுகம் செய்வது வழக்கம் என்றார். என் பெயர் மோகன், நான் ஒரு ஆர்கிடெக்ட் என்றார்; சகோதரர் சார்வாகன் என்றார். அவரை இந்திரா பார்த்தசாரதி கூட்டத்தில் பார்த்த நினைவிருக்கிறது என்றேன். திருநெல்வேலி சுற்றியுள்ள தென்பாண்டி தலங்களுக்கு அந்த ஆண்டு கலையுலா. ஒரு சில சொதப்பல்களாயினும் மகிழ்வோடு கலந்துகொண்டார். அடுத்த பல ஆண்டு கலையுலாவிலும் கலந்துகொண்டார். தமிழ் பாரம்பரிய அறக்கட்டளையின் நிகழ்வுகளில் மாதாமாதம் பார்க்கும் வாய்ப்பு கிடைத்தது. முகநூலில் நட்பாயினோம்.

கூவம் நதிக்கரையில் செஞ்சி பெருமாள் கோவில்

எழுத்தாளர் வெங்கடேஷ் ராமகிருஷ்ணன் தொடங்கிய கூவம் கலாச்சார வரைப்பட திட்டத்தில் கூவம் நதிக்கரை வரலாற்று சின்னங்களை தேடி பயணங்கள் தொடங்க, அதிலும் மோகன் ஹரிஹரன் பங்கு கொண்டார். நான் ஒரு மாதம் அமெரிக்க பயணம் செல்வதை கேட்டு, என் மகன் மருமகள் லாஸ் ஏஞ்ஜலீசில் உள்ளனர்; என் மகள் மருகன் பிலடல்பியாவில் உள்ளனர், அங்கே சென்றால் நிச்சயம் அவர்களை பார்க்கவும் என்று உபசரித்தார். ஆறு ஏழு மாதங்கள் பழக்கத்தில் அவருடைய நட்புணர்வும் விருந்தோம்பலும் வியக்கவைத்தது. லாஸ் ஏஞ்சலீசில் அவர் மகனை பார்க்க இயலவில்லை. ஆனால் அமெரிக்க தலைநகர் வாஷிங்க்டனுக்கு சென்று என் உறவினன் ராமாநுஜனுடன் ஒரு வாரம் நானும் என் தம்பி ஜயராமனும் தங்கினோம். பிலடெல்பியாவிற்கு சென்று பெஞ்சமின் பிராங்கிளின் அருங்காட்சியகம் பார்க்கவேண்டும் நியூ ஜெர்சிக்கு சென்று தாமஸ் எடிசனின் ஆராய்ச்சிசாலை பார்க்கவேண்டும் என்று எனக்கு அந்த சமீபத்தில் ஆர்வம் பெரிதும் வளந்திருந்தது. 1990களில் அமெரிக்காவில் வாழ்ந்த போதே கொஞ்சம் ஆசை; ஆனால அப்பொழுது விடுமுறை கிடைத்தால் உறவினர்களை பார்க்கும் ஆசையும் இயற்கை காட்சிகளை பார்க்கும் ஆசையும் அதிகம் இருந்ததால் அங்கே செல்லவில்லை. இந்தியாவுக்கு திரும்பிய பின் கலை இலக்கிய வரலாற்றின் மீது வந்த ஆர்வம் போல் அறிவியல் தொழில்நுட்ப வரலாறு மீதும் ஆர்வம் சிந்தை வனத்தில் அக்னிக்குஞ்சாய் தணல் எரிந்தது. பிலடெல்பியாவிக்கு ஆர்த்தி அருண் இல்லத்திற்கே அன்புடன் வரவழைத்தனர். அவர்கள் சுட்டி குழந்தைகளின் துடிப்போடு, ஒரு நாள் பெஞ்சமின் பிராங்கிளின் வாழ்ந்த இல்லத்தையும் அவருடை அறிவியல் படைப்புகளையும் பார்த்தோம்.

பிலடெல்பியா - 2015ல் ஆர்த்தி அருண் குழந்தைகளோடு நான், ஜெயராம், ராமானுஜன்

முதல் நாள் இரவு அவர்கள் இல்லத்தில் விருந்து. தந்தை மோகன் வரைந்த ஓவியங்கள் சுவர்களை அலங்கரித்தன! விசித்திரம் என்னவென்றால் திருவான்மியூரில் உள்ள மோகன் சித்ரா தம்பதியினர் இல்லத்துக்கு செல்வதற்கு முன்பே பிலடெல்பியாவில் உள்ள ஆர்த்தி அருண் இல்லதிற்கு நான் சென்றதே.

இந்தியாவிற்கு வந்த பின் தானும் தன் மனைவியும் எகிப்து சுற்றுலா பயணம்  செல்வதாக கூறினார். என்னையும் அழைத்தார். போக ஆசை தான். ஆனால் ஒரு மாத அமெரிக்கப் பயணத்திற்கு பின் என் பயண ஆவல் குறைந்திருந்தது. ஆனால் தயக்கத்துடன் அழைப்பை மறுத்துவிட்டேன். பெரிய இழப்பாக இன்று தோன்றுகிறது. கூவம் பயணங்களிலும் பின்னர் அடையாறு பயணங்களிலும் சந்தித்துக்கொண்டோம். சார்வாகனோடு ஒரு சந்திப்புக்கு அழைத்தார், அதற்கு செல்ல முடியவில்லை; மற்றொரு இழப்பு.

தான் பிறந்த ஆரணிஎன்ற ஊரைப்பற்றி நவம்பரில் த.ப.அ-வில் மோகன் ஹரிஹரன் ஒரு உரையாற்றினார். டிசம்பர் மாதம் நடந்த பேச்சுக்கச்சேரியில் கலந்து கொண்டு மிகவும் ரசித்தார். தமிழ் பாரம்பரிய அறக்கட்டளைக்கு ஒரு பெரிய தொகையை நன்கொடையாய் அளித்தார்.

ஆர்கிடெக்ட் ஸ்ரீநிவாசன் என்று தன் குருநாதரை பற்றி அடிக்கடி புகழ்ந்து பேசுவார். தமிழ்நாட்டில் பல சினிமா அரங்ககளை வடிவமைத்தவர் அந்த ஸ்ரீநிவாசன். சவீதா பலகலைகழகத்து நிறுவனர் மற்றும் சான்சிலர் வீரையன் அவர்களோடு மோகன் அவர்களுக்கு நாற்பது ஆண்டு நட்பு. வீரையனின் மகளே, டாக்டர் சவீதா. ஸ்ரீநிவாசனின் பொதுப்பணிக்கு சரியான அங்கீகாரம் கிடைக்கவில்லை என்று வருந்திய மோகன், சவீதா கல்வி நிறுவனங்கள் சாதனையாளர்களை கௌரவிக்க நடத்திய விழாவில், குருநாதர் ஸ்ரீநிவாசனுக்கு ஒரு சிறப்பு விருது வழங்க சிபாரிசு செய்தார். சான்சிலர் வீரையன் அந்த ஆசையை நிறைவேற்றி வைத்தார்.

அவர்கள் “தி பியூப்பில்” என்ற ஒரு பள்ளிக்கூடத்தையும் நடத்துகின்றனர். அதற்கு ஆர்க்கிடெக் மோகன் ஹரிஹரன் தான். அந்த பள்ளிக்கூடத்தை பார்க்க, அதன் கட்டுமானத்தின் பல பிம்பங்களை விளக்க ஒரு கட்டிட உலா நடத்தினார்; மிகச்சிறப்பான அனுபவம். அரைகளுக்குள்ளே வரும் இயற்கை வெளிச்சத்திலிருந்து, வெப்பத்தணிக்கைக்கு ஏற்ற செங்கல், குழந்தைகளின் மனம் கவரும் வண்ணங்கள், ஓவியங்கள் சித்திரங்கள், என்று பல சித்தாந்தங்கள் மிளிர்ந்த கலை. த.ப.அ மற்றும் அது போன்ற கலை மரபு பேச்சாளர்கள் சவீதா பள்ளி மாணவர்களிடம் பேசவேண்டும் என்று சிபாரிசு செய்தார். டாக்டர் சவீதா அவர்களே இதை நிறைவேற்றினார். பேராசிரியர் சாமிநாதன், பத்ரி சேஷாத்ரி, நான், மற்றும் பல கலை ஆர்வலர்கள்  பள்ளி மாணவர்களிடம் உரையாற்றினோம்.

பியூப்பில் பள்ளியில் மோகன் ஹரிஹரன் விளக்கம்

பியுப்பில் பள்ளியின் செங்கல் கட்டுமானமும்  நீச்சல் குளமும்

2017ல் பியுப்பில் பள்ளி ஆண்டு விழாவிற்கு என்னை சிறப்பு விருந்தினராக அழைத்தனர். நான் மயிலை சி.ஐ.டி காலனி தொடக்க பள்ளியில் நான்காம் வகுப்பில் படித்தப்போது என் பெற்றோர் சிறப்பு விருந்தினராய் அழைக்கப்பட்டனர். அந்த கௌரவம் எனக்கும் ஒரு நாள் கிடைக்கும் என்று நினைக்கவில்லை. சவீதா கல்லூரியோடும் இதனால் தொடர்பு ஏற்பட்டது. சவீதா பொறியியல் கல்லூரி இயக்குனர் மருத்துவர் ராஜேஷ், டாக்டர் சவீதாவின் கணவர். அவர்களது அழைப்பில் முதலாமாண்டு இஞ்சினியரிங் மாணவர்களின் ஓரியண்டேஷன் விழாவில் உரையாற்றினேன்.

இதற்குள் வராகமிகிரன் அறிவியல் மன்றம் தொடங்கியதால், கட்டுமான கலை பற்றி மோகன் அவர்களை உரையாற்ற அழைத்தோம். சிறப்பாக உரையாற்றினார். அவரது மற்றொரு மூத்த சகோதரர் கர்ணல் ரமணி ஹரிஹரன் கூட்டத்திற்கு வந்திருந்தார். ராணுவ உளவுத்தொழில் பற்றி  தான் ஒரு உரையாற்ற தயார் என்றார்; பின்னர் தமிழிலேயே உரையளித்தார். இளைய சகோதரர் சிவகுமார் ஹரிஹரன் த.ப.அ-வின் சாஞ்சி-மத்திய பிரதேச கலையுலாவில் கலந்துகொண்டார். சமீபத்தில் அவர் நிர்வகிக்கும் பாபாஜி பள்ளிக்கூடத்தில் என்னை பேச அழைத்தார். இப்படி அந்த குடும்பத்தோடு நீளமாக அகலமாக உறவு வளர்ந்தது.

வராகமிகிரன் அறிவியல் மன்றத்தில் நாங்கள் நடத்திய இந்திய வானியல் கணித வகுப்பிற்கு, டாக்டர் ராஜேஷ், டாக்டர் சவீதா, இரு மகள்களோடு வந்து சேர்ந்துகொண்டனர். ஆடிப்போய்விட்டேன். அடிக்கடி மயிலையில் நடக்கும் த.ப.அ நிகழ்வுகளுக்கும் கோட்டூர்புரத்தில் நடந்த வ.அ.ம நிகழ்வுகளுக்கும் அவர்கள் வருவதுண்டு. கல்வியிலும் கலையிலும் மரபிலும் அவர்களுக்குறிய ஆர்வம் அசாத்தியம். இவ்வார்வத்திற்கு மோகன் ஹரிஹரன் ஒரு முக்கிய தூண்டுகோள்.

சவீதா கல்லூரி வளாகத்தில் ஒரு முருகன் கோயில் கட்டினர். சிலைகளையும் மாடியில் உள்ள சிற்றாலயங்களையும் ஸ்தபதிகள் செய்தனர். ஆனால் கோயில் வளாகத்திற்கு மோகன் தான் ஆர்க்கிடெகட். கும்பாபிஷேகத்திற்கு முன் அவர் தலைமையில் ஒரு உலா நடத்தினார். அருகே கட்டிவந்த புதிய மருத்துவ வளாகமும் அவரது கைவண்ணம். அதையும் பார்த்து களித்தோம். 

கொரோனா ஊரடங்கில் மூடப்பட்ட கல்லூரிகள் வழக்கம்போல் ஆகஸ்டு மாதம் தொடங்காமல் டிசம்பர் மாதம் தொடங்கியபோது. இந்திய வானியல் கணித பாடத்தை ஒரு கோர்ஸாக அவர்கள் சவீதா பொறியியல் கல்லூரியில் நடத்த பேராசிரியர் மணிமாறன் வழியாக அழைப்பு விடுத்தனர். அவர்களின் வர்புறுத்தலில் இன்வென்ஷன்ஸ் அண்டு டிஸ்கவரீஸ் என்று ஒரு பாடத்தையும் இணையம் வழியாக இரண்டு செமஸ்டரும், 2021ல் ஊரடங்கு விலகிய பின் நேரே கல்லூரி வளாகம் சென்று நடத்தி வருகிறேன்.

இந்திரா பார்த்தசாரதி உரைக்கு வந்த மோகன் ஹரிஹரன் வழியாக வேலைவெட்டியாற்ற நான் மீண்டும் வாரம் ஒரு நாள் வேலை செய்யும் நிலைக்கு வருவேன் என்று கனவிலும் நினைத்துப் பார்க்கவில்லை.

மோகன் சித்ரா தம்பதியர் சதாபிஷேகம் - நண்பர்கள் உமையாள் சிவசங்கர் பாபுவுடன் நானும்

மோகன் அவர்களின் சித்திரங்கள்

சமீபத்தில் மோகன் சித்ராவின் மகன் பிருத்வி மகள் ஆர்த்தி மற்றும் குடும்பத்தினர் சென்னைக்கு வந்து அவர்களது சதாபிஷேகமும் ஐம்பதாண்டு கல்யாண வாழ்க்கையும் சேர்த்து கொண்டாடும் வகையில் ஒரு விழா நடத்தினர். என்னையும் தன் நெருங்கிய நண்பர்களில் ஒன்றாய் அழைத்தார்கள். வாழ்வில் அமைந்த பாக்கியங்களில் இவர்களது நட்பு, அன்பு, அக்கறை, உபசரிப்பு, ஆசிர்வாதம் எல்லாம் சேர்க்கை.

ஆகஸ்து முப்பது மோகன் ஹரிஹரனின் பிறந்தநாள்.

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