Astronomy and Mathematics in the Vedas

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 return. 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.

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

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This essay was first published in a series in Swarajya magazine

Related Links

My blogs on Astronomy and Mathematics

Number Notations in Sanskrit

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.

Katapayaadi in Devanagari letters

• 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

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This essay was first published in a series in Swarajya magazine

Related Links

My blogs on Astronomy and Mathematics

An Introduction to Indian Astronomy

Vedic Mathematics is a popular catchphrase. Vedic Architecture is another. Rarely have I heard the phrase Vedic Astronomy though. Which is odd. Aryabhata, famous for being a mathematician, was primarily an astronomer. Mathematics was a chapter in his astronomy book.

In fact, for several centuries, Mathematics was a chapter in Indian Astronomy books. Let me rephrase. Ganitam was a chapter in Jyotisham books. Isn't Jyotisha astrology or is it astronomy? Was it superstition or science or an inseparable mix of both?

Jyoti is the Sanskrit word for light. In this particular context, it means “celestial object that emits light”. Jyotisham is the science of such objects, which includes the Sun, the Moon, the planets, and stars. A jyotishaH studies these celestial objects.

We think of the telescope as the instrument that revolutionized astronomy. But astronomy was a science for several thousands of years before the telescope. All ancient culutres, Egypt, Sumeria, China, India practiced extensive astronomy long before the ancient Greeks, who adopted or acquired quite a bit of their astronomical knowledge from some these cultures. The most important tool for astronomy was not just the naked eye, but observation and recording.  The second most important tool was mathematics.

So entwined was mathematics with astronomy in India, that the word gaNakaH (mathematician) was effectively a synonym for astronomer.

Kaala Dik Desha Jnaanam

What was the point of studying these? Surely, to cast horoscopes, find a bride or groom, the right time to perform a wedding, build a house or palace or temple, start a journey, etc? That is the popular belief. I’ve rarely seen a dissenting opinion anywhere, especially from atheist scientists. The oldest surviving jyotisham texts we have, the Rg Vedanga Jyotisha and Yajur Vedanga Jyotisha very clearly explain the purpose of studying this field.

तस्माद् इदं कालविधानशास्त्रं यो ज्योतिषं वेद स वेद यज्ञान् ॥

tasmaad idam kaala-vidhaana-shaastram yo jyotisham veda sa veda yajnaan

 

"kaala-vidhaana-shaastram" means “the science of determining time ”. The second half of the sloka means : “He who knows jyotisham knows the yajnaas”. The primary purpose was to calculate time, so that the right yajnaas can be performed at the correct time.

Another work expands this definition to “Kaala-dik-desha-jnaanaartham” : To find time, direction and geography.

All personal uses of jyotisham were secondary.

Eras of Indian astronomy

We divide Indian history political into several eras, usually as Hindu era, Muslim era, British era, and post Independence. We can similarly divide the history of Indian Astronomy into several eras:

Eras of Indian Astronomy
---	Vedic Era
1400 BC	Vedanga Jyotisha
500BC - 500 AD	Rishi Siddhantas
500AD - 1500 AD	Classical Era

 

The only surviving text of the Vedic period are the Rg and Yajur Vedanga Jyotishas. They both have 36 identical slokas, but the Yajur version has one extra sloka. The author of both Vedanga Jyotishas  was Lagadha, of whom we know extremely little. He compiled that knowledge preceded him.

The Vedas themselves are about ritual, revelation, philosophy, prayer, etc. Their purpose is not astronomy or mathematics. Six additional subjects of study, called Vedaanga-s, were devised to help students learn, interpret and understand the Vedas. These Vedaangas are:

The Six Vedangas
Shiksha Phonetics   VyaakaraNa Grammar Chandas  Prosody        Nirukta         Etymology	Linguistics
Kalpa             Ritual        Jyotisha    Astronomy	Mathematics

The first four of these deal primarily with language (Linguistics) and the last two primariy with mathematics. It is worth observing, that the Sangam Tamil book Tolkappiyam deals with the first three subjects, for Tamil, as three different sections.

A beautiful sloka describes the importance give to Jyoisha

यथा शिखा मयुराणां नागानां मणयो यथा ।
तद्वद् वेदाङ्गशास्त्राणां ज्योतिषं मूर्धनि स्थितम् ॥ 

yathaa shikhaa mayurANAm naagaanaam maNayo yathaa
tadvad vedaanga shAstrANAm jyotisham murdhani sthitam 

Yathaa - Just as
shikhaa - crest
mayurANAm – among peacocks
naagaanaam – among serpents
maNayo - jewels
tadvad – thus too
vedaanga  shAstrANAm – among vedanga shastras
jyotisham – astronomy
murdhani – on the head
sthitam – stands 

Just like crests on peacocks, and jewel on serpents
Thus among vedanga shaastras, Jyotisham crowns the head 

Sulba sutras

Texts called the Sulba sutras are part of Kalpa, and they describe the geometry of Vedic altars. Altars for the yajnas were to be constructed in various shapes, including circle, semicircle, square, octagon and compex figures like birds. The bird shaped altar consisted of several hundred rectangles, trapezeums, triangles, etc. and constructing them with bricks of predetermined size was a complex geometrical exercise. These were measured using ropes (sulba or rajju). This is one noticeable practical difference between the practical origins of Indian versus Egyptian or Greek geometry. In the latter culutres, the primary practical motivation was measurement of land, especially farmland. Hence the subject was named geo (earth) metry (measurement). This is also a major difference between Vedic and Jain or Buddhist mathematics; since the latter religions did not need performance of yajnas, the theoretical pursuit of scholars of those faiths were also somewhat different.

There were several Sulba sutras, authored by different people and named after them like Baudhaayana, Aapastambha, Kaatyaayana etc.

But us let get back to astronomy now. And the determination of time.

Solar, Lunar, Stellar Days

The period from one sunrise to another, is called a dina, divasa or aha in Sanskrit. Since the Sun causes the day, he is called Dina-kara (the maker of the Day). The English astronomical term for this is Solar Day.

The period from one moonrise to another, is called a thithi in Sanskrit. (Lunar Day).

There are thirty thithi-s, fifteen from new moon to full moon(called waxing fortnight in English and shukla paksha in Sanskrit), and fifteen from full moon to new moon (called waning fortnight inEnglish and krishna paksha in Sanskrit). This thirty thithi period is maasa. Note that each thithi or lunar day is slightly shorter than each solar day. Twelve lunar maasa-s will total upto 354 rather than 365 solar days. This quickly led to a mismatch between lunar year of  and solar years, and was resolved using a concept called adhika maasa (or extra month). Once every five years, adhika maasas were used to realign solar and lunar years. According to Vedanga jyotisha, this five year period was called yuga. But in all later siddhantas, the yuga is a much longer period of several lakh years.

We can see about six thousand stars in the night sky, but Indian astronomy has a named cycle of twenty seven. Why? There is a very scientific reason. These twenty seven stars lie along the orbit of the moon around the earth.

Each night the moon changes its position in the sky against the background stars (because it is orbiting the earth). Oriental historians called these lunar mansions. The name of the brightest star next to the moon on each night was observed to fall in a certain cylce…. Ashwini, Bharani, Kriththika, Rohini….with series ending in Revathy on the twenty seventh night. So each night or day was called by that star… or nakshatram; the English astronomoical term for this is Stellar Day. They also noted that moon became  a full moon, only when it was next to some of these twenty seven; hence the months were named after those stars. So the month where the full moon was next to Chitra was called Chaitra; when next to Visaka was called Vaisaaki, when next to Kritthikaa was called Kaarthika, when next to Mrigasirsha the month was called Maargasirsha….and so on. You can see local adaptations of these names in various languages. So Vaisakhaa is called Baisaaki in some languages, Sravishtam is called Avittam in Tamil, etc.

Notice that this system of individual stars, contrasts starkly with the twelve Sumerian/Greek constellations called the zodiac

Divisions of the Day

The sun moon and the stars form natural clocks in the sky. These were useful to divide years into days and months, but the days themselve were divided into naadi and vinaadi using artificial devices like water pots of particular volume. These terms evolved over the ages.

A verse by Aryabhata concisely explains these divisions

वर्ष द्वादश मासास्त्रिंशद्दिवसो भवेत् स मासस्तु ।
षष्टिर्नाड्यो दिवसः षष्टिश्च विनाडिका नाडी ॥

varsha dvaadasha mAsA   trimshad divasO bavEd sa mAsa tu
shashTi naaDIya divasa    shashti vinaaDikaa naaDee 

varsha - Year
dvaadasha - twelve
mAsA   - months
trimshad - thirty
divasO - days
bavEd - become
shashTi – sixty 

A year is twelve months, a month thirty days
A day is sixty naadi-s, a naadi is sixty vinaadi-s 

Aryabhata is estimated to have lived nearly two thousand years after Lagadha, the most ancient Indian astronomer; what a pity most of us have never heard of him. In fact, we are chronologically closer to Aryabhata than he was to Lagadha. How did the science develop in those two millennia? We will discuss these next.

References

• 1.      Facets of Indian Astronomy, KV Sarma, 1975
• 2.      The Aryabhatiya of Aryabhata, KV Sarma and KS Shukla
• 3.      Oral Traditions of Sanskrit, Powerpoint presentations of Prof S Swaminathan

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This essay was first published in a series in Swarajya magazine

Related Links

My blogs on Astronomy and Mathematics

Video of my Lecture at Indian Science Festival, Pune: 

The men who knew Zero to infinity January 2020

The Languages of Science

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

A Sculptor's muse

Softly O softly she floated ashore
On the waves of his dream to his mind's very core
Like the dew on the grass from the soul of the breeze
In the still of the night with the greatest of ease

Smoothly O smoothly he etched in her shape
For devas to gasp and manushas to gape
Firmly O firmly he hammered the stone
'Til rock came alive as flesh blood and bone

Her lips promised love, her breasts spoke in rhyme
Her hips gently prodded her anklets to chime
She smiled with her eyes and laughed with her heart
As he breathed into her the essence of art

On the return from Jhansi after January 2025 THT Site Seminar in northern Madhya Pradesh, this sculpture popularly known as Gyaraspur lady, now kept in the Gujari Mahal museum in Gwalior fort, inspired this poem above. "Gyaraspur lady" is the sculpture of a shaala-bhanjika from a place called Gyaraspur near Vidisha, which is south of Gwalior. In 2019, THT site seminar was conducted in areas around Bhopal including Vidisha and Gyaraspur. That's when we came to know about it, when Vallabha Srinivasan gave a preparatory lecture about it. Mr Venkatesiah, retired Regional Director of ASI, accompanied us as an expert, and informed us that this statue was under high security, behind a barricade, because of several attempts at stealing it. While some were waxing eloquent about the grace, beauty, allure, posture and other aspects of this sculpture, I was reminded of an apsara sculpture from Nageshvaran temple in Kumbakonam, Tamilnadu - one of four such beautiful apsara figures in that temple. This particular apsara, west of the Dakshinamurthy shrine, is soft, supple, demure, enchanting. When I mentioned this, Sowparnika, who was nearby showed it from her photo collection in her phone. The Gyaraspur lady photo is by Suresh Priyan, who was also on the Site Seminar.

Phrases and the tune of Sarojini Naidu's poem, "The Palanquin Bearers", which I had read in school, and loved for the very Indian sounding chandas/yaappu (prosody) of this English poem, floated into my thoughts. On the return journey by train on January 27, I wrote this poem - it applies to both sculptures and the shilpis who sculpted them.

Gyaraspur lady - photo by Suresh Priyan

Apsara at Kumbakonam Nageshvaran temple - photo by Sowparnika

Related Essays in my Blog

மல்லை சிற்பியர் வாழ்த்து

My poems

My Essays on Art