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.

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

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