Aryabhata

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.

Related Links

My blogs on Astronomy and Mathematics

The Kerala School of Mathematics and the Modern Era

The Classical Era Continues

For centuries after Bhaskara, the classical era continued. The Ganita Kaumudi of Narayana Pandita is a fascinating book, with an interesting structure and several novelties. Like Mahavira, he gave both rules and a series of examples. He expanded Arybhata’s formula for sum of series of numbers, their squares, and cubes, to any arbitrary power.

Effectively, he provided the formula

(S_n)^r =  [n(n+1)(n+2)…(n+r)]  / [1.2….(r+1)]   

Narayana also expanded on the meru prastaras (Pascal’s triangles) of Pingala and Hemachandra into more elaborate versions. Permuatations, combinations, areas and volumes of more compelex geometric shapes were other areas of exploration and explanation.

His famous novelty was magic squares, squares filled with numbers so that each row and column adds up to the same number. He explained procedures to create magic squares of desired sizes, how to fill them up for a given total, and certain fascinating variations, including use of negative numbers. He extended this to magic rectangles, and other figures, like squares within squares, triangles within triangles etc. He also devised many more such shapes like vajra (diamond), padma (lotus), padmavrtta(lotus circle), manDapa(canopy), sadashra (hexagon), dvaadashaashra (dodacehedron) etc filled with such numbers. These are explorations with perhaps no practical applications, especially in astronomy.

Later Jaina mathematicians Dharamanandana and Sundarasuri, continued explorations on magic squares and similar arrangements.

But from the fourteenth century, Kerala became the locus of several new siddhaantas, bhaashyas, karanas etc., that it is now called the Kerala school of mathematics. 

The Kerala school

Shortly after Aryabhata, a mathematician called Haridatta had composed a work title ParahitaGanita based on Aryabhatiyam. After that we seem to have a vaccum in Kerala until Madhava (c 1340 AD) of Sangamagrama (Kudalur in Malayalam). What followed is an astonishing continuity of the guru-shishya parampara from Govinda Bhattatiri to Rajaraja Varma.

Madhava was famous as a skilled instrument maker. Indian historians of mathematics consider him the pitamaha of the Kerala school. Perhaps his most famous contribution is sums of infinite series. To calculate the circumference (paridhi) of a circle, said Madhava, we must multiply the diameter (vyaasa) by four times one minus tri-sharaa-aadi-vishama-samkhyaa-bhaktam-rNam-svam-prtak. A phrase of compactness, which Aryabhata would have enjoyed. In other words, a sequence of odd (vishama samkhyaa) denominators (bhaktam) starting (aadi) with three (tri) and five (sharaa), negative(rNam) svam(positive) alternately(prtak). The word sharaa here is a bhutasamkhya (see third essay) word for the five arrows of Manmatha, whose archery takes precedence over Rama and Arjuna and even Tripurantaka, in this case.

     paridhi = 4 * vyaasa * (1 -1/3 + 1/5 - 1/7 + 1/9 - 1/11....)

The sum of the series in the brackets adds up to π/4, and is famous as the Gregory Leibniz series.

Madhava’s series was quoted and a proof (upapatti) also given a century later by Jyeshtadeva.

Parameshvara – Like Brahmagupta providing a sphuTam to Paitamaha Siddhaanta, Parameshvara observed that over time predictions of earlier astronomers did not agree with observed positions based on calculations. In such a situation, he observed, one must adjust one’s methods and caluclations, because planets and stars will confirm to He titled his book Drg Ganitam. This title, which means Observed Calculations,  is a popular phrase for jyotisham in south India, though the author himself has faded from public memory. His shishya NilakanTha referred to him as Paschimaam Bodhi, the western scholar.

Nilakantha Somayaaji – More unknown than even Brahmagupta, NilakanTha was a polymath like Varahamihira. He was a scholar of Shad darshana (the six philosophies of Hinduism, and also in vyaakarana, chandas, the Bhagavata and various such litereature.  He also studied Vedanga Jyotisha, Pancha Siddhaantika, Brhat Samhita etc. This historical curiosity and scholarship may have shined in other scholars, too, but in Nilakantha we have contemporary evidence. He was also a prolific composer, of several texts.

He was a friend a Sundararaja, a jyotisha of the neighboring Tamilnadu, and took the effort to compile a written list of answers for questions posed by the former, compiled into a book called Sundararaja Prashnottara. Aficionados of European science may be reminded of the extensive correspondonce of Franklin, Newton, Darwin, Humboldt etc.

A ninth century mathematician called Virasena in his commentary Dhavala gave this equation, that the sum of all powers of 1/4 is 1/3. One sees the reflection of this in Madhava’s several infinte series. Nilakantha questioned this apparent absurdity. How does the sum of this infinite series increase to that finite value(1/3), and that it reaches finite value?

He reasoned and explained it by deriving the following sequence of results

As we add more terms, argued Nilakantha, the difference between 1/3 and the powers of 1/4 become extremely small, but never zero, unless we add terms upto infinity. In the 20^th century, Ramanujan reveled in such series.

Quasi Heliocentric theory NilakanTha’s questioning of an assumption of planets’ latitudes(vikshepa), and his subsequent discovery was truly astronomical, pardon the pun. From the siddhantas through Bhaskaracharya, all astronomers used a slightly different method to calculate the latitudes of Mercury (Budha) and Venus (Shukra), than they did for the other planets. This niggled most of them, as “inappropriate”, especially Bhaskara, who then consoled himself with Prthudaka Svami’s exlpanation. But NilakanTha questioned this acceptance, and modified his computation, and effectively the orbital model for these planets. He came to the conclusion that these two planets revolve around the Sun (but in his model, the Sun still revolves around the Earth). The geometrical argument is too complicated not only for this article, but even perhaps for those who are not astronomers, so I will only present a visual illustration of his model here.

Effectively, NilakanTha formulated a quasi-heliocentric theory : sun centric for inner planets, and earth centric for everything else.

Jyeshta Deva a junior of Nilakantha wrote a work called Yuktibhaasha, in Malayalam, not Sanskrit. He extensively dealt with several proofs and also developed the theory of infinetesimals to greater refinement and sophistication.

Modern Era

The modern era was introduced gradually into India, with the advent of the East India company, and the influx of European scientific discoveries it brought along. I’ll refrain from the delving into the astronomical the Europeans themselves conducted here, including the establishment of the Madras observatory, the Madras meridian, observations of Venus transit, the discovery of Helium etc.

Sawai Jaisingh’s several Jantar Mantar are perhaps the most famous observatories in India, and really belong to the modern era. He was also instrumental in commissioning translations of Euclid and Ptolemy from Persian.

The most fascinating ganaka or siddhanta acharya of the modern era was Pathani Samanta Chandrasekhar, of Orissa. Utterly unaware of the European advances in astronomy including telescopes, electricity, gravity and optics, or even of the Madras observatory, and apparently not even aware of the Kerala school, he independently discovered the quasi helio centric theory of NilakanTha Somayaji. He also made several corrections and improvements to Bhaskara’s Siddhanata, using only the naked eye, besides develiping his own new instruments. His book Siddhanta Darpana, with 2500 slokas was published by PC Ray . He is only jyotishaacharya of whom we have a genuine portrait picture. The Bhubaneshvar obseratory is named after him.

Orientalists

Sir William Jones, judge of the Supreme court of Bengal, founder of the Asiatic Society of Bengal, discoverer of the Indo-European family of languages, translator of Manu Smriti and Shakuntalam into Sanskrit, rediscoverer of Pataliputra, ad infinitum, also declared that India gave the world algebra. Why we learn in school about FaHien and Hiuen Tsang, al Beruni and ibn Batuta, but nothing of this giant of Indian scholarship, I’ll never know. Some of us Indians possess the same myopic prejudice against such European stalwarts as the haughty colonialists did of us lesser Indians. It may also be cognitive dissonance. 

The Asiatic Society and such similar institutions like the Madras Literary Society produced extraordinary scholarship about several aspects of Indian history and culture, including astronomy and mathematics. Henry Thomas Colebrooke, George Thibaut, John Playfair, Burnell, Burgess, and such other European scholars translated, analyzed and compiled the history of Indian sciences and exposed them to the world., Charles Whish, and his collaborators John Warren and George Hyne played a significant role in bringing to light the Kerala school in the early nineteenth century. This was also the period when English effectively replaced Sanskrit as the primary language of science in India. The spectacular advances brought about by the Industrial revolution, and subsequent scientific advances in Europe, and the corresponding rise of European intellectual and military supremacy, drowned out the accomplishments of Indians in the past (and other cultures as well). The rest of the world is catching up in application of technology, but is stall far behind in the curiosity and spirit that drove discoveries and inventions. That is now changing. 

European orientalists did not work in isolation. Indian pandits, like Sudhakara Dwivedi, Chidambaram Iyer, and numerous others helped translate Sanskrit texts into English, French and other languages. 

Sir William Jones

In the twentieth century scholars like Bibhutibhushan Dutta and Avdesh Narayan Singh, TS Kuppanna Sastri, KV Sarma, KS Shukla, Rangacharya and others have translated books, and given us various detailed studies of such works also. Unfortuntately, these remain confined to the small community of scientists who know Sanskrit.

KV Sarma

The future

In my first essay in these series, I presented a quiz, contrasting our ignorance of Indian scientists and their contributions versus our knowledge of European scientists. I hope some of these essays have addressed that lacuna: we know a little about Brahmagupta and Nilakantha. There was much science in the investigations of our star gazers, than taught of in our philosophy. I hope that it has provoked curiosity about Indian history, especially of science and technology, and a desire to at least seek some of this information, of which we have zero exposure in the public education system. I hope also that we as a society, try to learn about sciences of various ancient cultures, such as China, Egypt, Olmecs-Mayans, Sumeria, etc. It is fascinating that Mayans independently discovered the zero, but not the place value system, for example. And our 24 hour clock is based on Egyptian astronomy.

I am not a mathematician, an astronomer, or a historian; I don’t know Sanskrit, except for a smattering of vocabulary, especially technical and scientific. If even I can learn enough to write a series like this, what can you not do?

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This essay was first published in Swarajya magazine

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

References

1Facets of Indian astronomy, KV Sarma

2 Studies in Indian Mathematics and Astronomy (Selected articles of Kripa Shankar Shukla) – Kolachana, Mahesh, Ramasubramanian

500 Years of TantraSangraha, MS Sriram, MD Srinivas, K Ramasubramaniam, 2002

Lecture on Samanta Chandrasekhar, by Dr Prahlad Chandra Naik , Seminar in Univ of Madras, February 2011.

The Discovery of Madhava series by Whish, Ganita Bharati Vol 32, 2002

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

Related Links

My blogs on Astronomy and Mathematics

From Brahmagupta to Bhaskara

 India’s greatest Mathematician

If there were a contest for who was India’s greatest mathematician, who would win? But first, who would be the candidates? Srinivasa Ramanujan? Aryabhata? How about Bhaskaracharya? Why not reach far back into Vedic times? Do Baudhayana or Apastamaba qualify? Should it be a public opinion poll? Does one have to be a great mathematician himself to judge?

Did any Indian jyotisha ever raise this question? Yes! Bhaskara, the 12^th century mathematician famous for Lilavati and Siddhanta Shironmani, himself considered the greatest by some mathematicians, gave such a title – Ganaka Chakra Chudamani. Not to Aryabhata or Baudhayana, but to Brahmagupta.

Brahmagupta who?! What did he do? When and where did he live? Does he have an ISRO satellite named after him? Or even a bus stand?

If there is one severely neglected, least translated, often maligned, and barely known mathematician, perhaps mathematician, in Indian history, it must be Brahmagupta. Perhaps it is because he used harsh language himself against Aryabhata, Srishena, Varahamihira and Vishnuchandra among others.

The most fervent criticism of Brahmagupta, often by self proclaimed rationalists, is that he used orthodoxy to subvert the revolutionary concepts brought out by a more logical Aryabhata. These often stem from the false belief that a "secular and heterodox” Aryabhata proposed a heliocentric theory, and that somehow Brahmagupta derailed this with orthodoxy.

ब्रह्मोक्तं गृहगणितं महता कालेन यत् खिलीभूतम् ।
अभिधीयते स्फुटं तत् जिष्णुसुत ब्रह्मगुप्तेन ॥ 1 ॥

Brahma-uktam gruha-ganitam mahataa kaalena yat khilibhootam
abhidheeyate sphuTam tat jishnu-sutam brahmaguptena 

Translation The planetary calculations (gruha-ganitam) explained (uktam) by Brahma have decayed (khilibootam) because of passage of a long (mahataa) time(kaala). Hence a correction (sphuTam) is presented (abhidheeyate) by Brahmagupta, son (sutam) of Jishnu.

This is the introductory verse in his work BraahmaSphutaSiddhaanta. I leave it to the reader to judge the orthodoxy of someone who announces that he offers a full book as a correction to the siddhanta of the Creator Brahma himself, because it has decayed with time.

The BraahmaSphutaSiddhaanta is a massive book, with 1008 stanzas arranged in 24 chapters. It became the role model for most later siddhaantas. So big was were these siddhaantas, that smaller texts called karanas and tantras were prepared for the regular use. Brahmagupta himself prepared such a karana, called  Khandakaadhyaka. 

Brahmagupta advocated observation, and use of instruments. He dedicated an entire chapter to yantras, or simple devices to observe the sky. Most of these devices were made of wood, clay and such perishable material, not complicated instruments with gears and levers and large metal bases that we associate with European astronomy after Galileo. Hence we lack even significant artifacts of the this science. Several of his observations were improvements over earlier astronomers, including Aryabhata.

Brahmagupta’s Innovations

But his innovations in mathematics are what evoked Bhaskara’s admiration. Brahmagupta gave us the mathematics of zero. Its addition, subtraction, multiplication and division. He explained negative numbers (which were written with a dot over them) and their properties. God may have created integers; Brahmagupta explained them.

 

We admire the giants of complexity like Gauss, Euler and Newton, more than the geniuses whose simple inventions wrought about giant leaps in ease. For example, the basic arithmetic signs we use were invented only just before Newton.

A Bija, by any other name

Today we call unknowns “variables” and use English letters like x,y,z or Greek letters like theta, delta, omega to represent them. But the names of unknowns varied over centuries.

 

We also call a,b etc coefficients in an equation such as ax²+bx=c. Indian algebra perhaps suffered, never successfully coining a single term for this. (To be fair, nothing in Sanskrit ever seems to possess a unique name. But fortunately, unlike Mahavishnu, most concepts have less than a thousand names). Brahmagupta himself referred to coefficients as samkhya (number) or gunaakara (multiplier). His commentator, Prthudaka Svami called it anka or prakriti.

 

Brahmagupta invented the equation (without these European operator symbols). He called it sama or samikarana. The two sides were calle pakshas, itara paksha and the apara paksha, one written above the other. He devised logical names for exponential powers above four, as pancha-gata, shad-gata etc. rather than isolated names like varga, ghana for square and cubes. With this insight, he also realized that only coefficients of like powers, which he called samaana-jaati, could be added. He arranged equations so that like powers lined up. This picture shows Prthudaka Svami’s format, based on Brahmagupta’s notation.

 

What Lavoisier did for elements, and Linnaeus for species, Brahmagupta did for algebra. 

Brahmagupta also came up with a general algorithm for solving quadratic equations, dealt with fractions of various types, cyclic quadrilaterals, used second order differences for sine calculations, and explored integer solutions for x,y for second order indeterminate equations of the type x²-by² = k. He deviced a method called bhaavana for solving these.

Improvements and Commentaries

A number of jyotishas thrived in the centuries following Brahmagupta. No single book or school of astronomy was dominant or exclusive in India. Aryabhateeyam, BrahmSphutaSiddhanta and SuryaSiddhanta were used in parallel for several centuries, Bhaskara’s Siddantha Shiromani was based on BrahmaSphutaSiddhanta. The SuryaSiddhanta predating Varahamihira,  and the most popular siddhanta of India, was anonymously updated with several concepts discovered by Brahmagupta.

 

 

The classical era was one of thriving innovation, and saw an explosion of manuja grantham : siddhantas, bhaashyaas, vartikaas (explanations of commentaries), karanas, tantraa, and novelties like vaakya-panchaangas, a surprising number of which have been preserved, edited, published and some even translated into English in the last few centuries. Criticism, correction, observation, refinement, innovation marked this period of several centuries and across various geographies.

An interesting aside for an economist, is the variety of currencies and coins (dinara, paNaa, kaarshapaaNa, puraaNa, svarNaa) and weights (pala, krosha) and measures (angula, hasta) discussed in the various books. The primary focus is on astronomy, but every siddhanta discusses principal, interest, compounding, rate of growth, and such monetary calculations also.

Mahavira

Mahavira, the Jain mathematician who composed Ganita Saara Sangraha wrote the first mathematics book, shorn of astronomy. The structure of his book is that first two or three stanzas in each chapter explain an algorithm or formula, and the rest of the stanzas are problems of that type to be solved by the reader. His use of Jaina symbols, temples, methods of worship, calculations etc. are singular hallmarks of the book. Mahavira revels in several types of fractions:  bhaaga (simple fraction), prabhaaga( fractions of fractions), bhaagaabhaaga (complex fractions), and so on. For example, one problem posed is below:

दिवसैसत्रिभिस्सपादैरयोजनषट्कं चतुर्थभागोनम्
गच्छति यः पुरुशोसौ दिनयुतवर्षेण किं कथय ॥ ३ ॥

divasais-tribhis-sa-paadair-yojana-shaTkam caturta-bhaaga-unam
Gacchati yaH purusho-asau dina-yuta-varsheNa kim kathaya

Translation The man (purusha) who (asau) walks (gacchati) quarter (caturtha-bhaaga) less (unam) than six (shaTka) yojanaas in three (tribhi) and quarter (paadai) days (divasau), tell (kataya) how much (kim) he walks in a day (dina) and (yuta) a year (varsha). 

Bhaskaracharya

The Lilavati of Bhaskara, author also of Siddhanta Siromani, is famous even to those unfamiliar with mathematics, as an example of beautiful poetry, and has a popular legend around it. Like Mahavira, Bhaskara tossed in several examples from daily life to pose mathematics problems, and like Varahamihira, he reveled in his poetic talents. Lilavati is the usually only mathematics book that Sanskrit dictionaries quote. It inspired innumerable commentaries, over centuries, translation into multiple languages and became the standard textbook of Indian mathematics.

Bhaskara corrected Aryabhata’s wrong formula for the volume of a sphere, which escaped even Brahmagupta (who corrected Aryabhata’s wrong formula for volume of a tetrahedron).  He also gave correct volumes for surface area of a sphere. His metaphor of a net covering a ball (kandukasya jaalam), for sphere volume hints that he had stumbled upon the germ of the idea of infinetismals and calculus. But these fields would only develop later centuries, in Kerala.

Bhaskara also introduced the concept of kha-hara (a number divided by zero) for infinity (not just the philosophical ananta (endless).

Bhaskara was also the among the earliest to provide proofs of some of his derivations, and not leave it to commentators, or only teach students. After brief explorations by Pingala and Varahamihira, Bhaskara also explored permutations and combinations.

By Bhaskara’s time, algebra had developed into an advanced state. He acknowledges that he built on the works of his predecessors Sridhara and Padmanabha.

Historical perspective

Indian mathematicians were using irrational square roots for a thousand years and sines and cosines for several centuries before discovering negative numbers. The inspiration for negative numbers comes from commerce and the notion of debt, not any religious philosophy. It needed six centuries and a Bhaskara to correct Aryabhata’s sphere volume mistake. Bhaskara realized division by zero yields infinity, but didn’t fully grasp its consequences.

From the finite series of Aryabhata to the infinite series of Virasena took only two centuries. They discovered infinite series summed up to a finite number for six centuries before questioning it.

Just as the steam engine was invented a century before the much simpler bicycle, the history of mathematics is replete with examples of complex concepts being discovered before much simpler concepts. Astronomy inspired extraordinary mathematics, but also frequently fooled and misled the greatest of mathematicians.

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This essay was first published in a series in Swarajya magazine

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

References

BrahmaSphutaSiddhanta 1966 edited by RamSwarup Sharma; Introduction by Dr Satyaprakash

  History of Hindu mathematics, Bibhutibhushan Dutta and Avdesh Narayan Singh, 1935

4Ganita Saara Sangraha, translation by Prof Rangacharya, Presidency college, Madras, 1912

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

Related Links

My blogs on Astronomy and Mathematics

3.    

Era of Rishi Siddhantas

 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