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.    

A Sanskrit anthem for Mathematics

தமிழில் இந்த கட்டுரை

In the ninth century, most of today’s Karnataka was ruled by a king of the Rashtrakuta dynasty called Amogavarsha Nrpatunga. One of his inscriptions says that his kingdom extended from the Godavari to the Kaveri, so his territory was not confined to modern Karnataka.This king was also a very special scholar – he composed the oldest surviving great literature in the Kannada language, Kavirajamarga. 

In his court lived a great mathematician, Mahavira, who composed a book titled GanitaSaraSangraha. This is as remarkable as Kavirajamarga, because this is the oldest book in Sanskrit,exclusively dealing with mathematics. Hundreds of books  were composed in previous centuries in Sanskrit, including the famous books of Apastambha, Baudhayana, Aryabhata, VarahaMihira and Brahmagupta. But those books were either sulba sutras or jyotisha sutras, and mathmatics was one of the components of those books. 

GanitaSaaraSangraha is the first book where mathematics is the primary subject. Several later astronomers like Madhava, Parameshvara, Bhaskara and Nilakanta Somayyaji wrote books on Astronomy, which had chapters on Mathematics, so the older practice did not fade out.

Prof Rangacharya of Presidency College, Madras translated GanitaSaraSangraha into English in 1912. The book was translated first into Telugu by Pavuluri Mallana, in the eleventh century and most recently in 2000, to Kannada by Prof Padmavathamma, of the University of Mysore. To my knowledge, there is no Tamil translation, except perhaps my own translation of a mere three stanzas.

Mahavira, was a Jain. The first stanza of the GanitaSaraSangraha has the phrase namastasmai jInendrAya mahAvirAya नमस्तसमै जिनेन्द्राय महावीराय (Salutations to Jinendra Mahavira), a reference to the last Jain tirthankara, Vardhamana Mahavira. Interestingly, in their first slokas the Hindu astronomers Aryabhata salutes Brahma, Brahmagupta salutes Siva, and Nilakanta Somayyaji salutes Vishnu. An interesting  comparison may also be made to the famous Meguti inscription of Chalukya king Pulikesi in Aihole, which begins with a similar salutation to Jinendra (Jayati Bhagavaan JinendraH जयति भगवान् जिनेन्द्रः) 

This is the other remarkable aspect, because the book is in Sanskrit, not Prakrit. Popular belief is that most works by Jains and Buddhists were written in Prakrits (Jain works in Ardha Magadhi, and Buddhist works in Pali). While this is true of several Jain and Buddhist compositions of the first few hundred years, not just for philosophy or religion, but also for sciences, later Buddhists and Jains wrote in Sanskrit, which became the lingua franca not just for people who followed the Vedic religion, but also for the sciences and the arts. 

This is an extensive topic, which is not well-known to learned Indians, and I wont discuss it here. But this is very similar to how Latin was used in Europe, after the fifteenth century, not just as the language of the Christian clergy, but also of science and arts. Hence, Linnaeus developed Latin nomenclature for naming plants and animals by genus-species, Newton wrote his book book on physics Principia Mathematica de Naturalis in Latin, chemists from Lavoisier onwards, used Latin words to name most of the elements. And since the laws of Europe use Roman law to guide them, which thence guide the Constitution and laws of former European colonies like India, Pakistan, USA, Australia, most of Africa and the Americas, Latin is the primary language of global law, except in Islamic countries and China.

But in this blog, I wont discuss the new mathematical concepts expounded by Mahavira, but merely quote three stanzas of his poem, which I believe should be declared the Anthem of Mathematics, and included in every mathematics school text book, not just in India, but in every culture broadminded enough to enjoy and agree with this poem. Here is the Sanskrit source and my English translation. My Tamil translation is here. 

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

laukikE vaidikE vaapi tathA sAmAyikepi yaH

vyApArastatra sarvatra sankhyAnam upayujyate

kAmatantre artatantre ca gAndarve nAtakepi vA

sUpashAstre tatA vaidye vAstu vidyAdi vastushu

cando alankAra kAvyeshu tarka vyAkranAdishu

kalA guneshu sarveshu prastutam ganitam param

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.

You can listen to this anthem rendered as a song by Diwakara Tanujaha here 

Meaning by words

laukikE – In worldly life

vaidikE – In Vedic learning

vaapi – also

tathA – likewise

sAmAyikepi – in religious practice

vyApAraH – in business

tatra – there

sarvatra – in everything

sankhyAnam – mathematics

upayujyate – is useful

kAmatantre – in romance

artatantre –in economics

ca – and

gAndarve – in music, the art of gandarvAs

nAtake – in dance and drama

api vA - also

sUpashAstre – in cooking

vaidye – in medicine

vAstu vidyAdi vastushu – in architecture and construction

cando alankAra – in prosody and poetry

kAvyeshu – in epic poetry

tarka – logic

vyAkrana- grammar

Adishu – in these

kalA guneshu – in arts

sarveshu – in everything

prastutam – is established (reigns)

ganitam – mathematics

param – supreme

If you liked this essay, you may enjoy reading these related blogs:

0. My Essays on Astronomy

1. Video of Prof MS Sriram lecture on GanitaSaaraSangraha
2. The Peacock’s Tail A interesting blog on GanitaSaraSangraha
3. Field's medallist Manjul Bhargava on Sanskrit and Mathematics
4. Some astronomy/mathematics slokas in Sanskrit
5. Nilakantha Somayyaji’s Sanskrit mathematical pun
6. Atyantakama Pallava’s poem (also sung by Sudharsanam)
7. Aryabhata’s sloka for pi
8. Varahamihira’s salutation to Agastya
9. My essay on Aryabhata in The Week
10. A comparison of timelines - Tamilnadu and Karnataka

ஸமஸ்கிருதம் புரிகிறதே!

மஹாவீரர் என்ற கணிதமேதை, ஒன்பதாம் நூற்றாண்டில், அமோகவர்ஷ நிருபதுங்கன் என்ற அரசனின் (இன்றைய கர்நாடகத்தில்) அவையில் இருந்தார். கணித ஸார ஸ்ங்கரஹம் என்ற நூலை இயற்றினார். இதுவே ஸமஸ்கிருதத்தில் முதல் கணித நூல். இதற்கு முன் ஜோதிட நூல்களின் ஒரு அத்தியாயமாகவே கணிதம் இடம்பெற்றது. சமீபத்தில் (ஆகஸ்ட் 31)  “இந்திய விண்ணியல் – வேத காலம் முதல் 16ஆம் நூற்றாண்டு வரை” என்ற தலைப்பில் சென்னை ஆழ்வார்பேட்டையில் ப்ரெயின்ஸ் ட்ரஸ்ட் குழுவிற்கு நான் உரையாற்றினேன். இன்னூலில் வரும் கவிதையின் ஒரு பகுதியை படித்து விளக்கினேன். உரை முடிவில் ஓரிருவர் ”எனக்கு ஸமஸ்கிருதம் தெரியாத போதும், இது மட்டும் புறிகிறதே” என்றனர். உங்களுக்கு எப்படி? என் மொழியாக்கம் கீழே.

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

லௌகிகே வைதிகே வாபி ததா ஸாமாயிகேபி ய

வியாபாரஸ்தத்ர் ஸர்வத்ர ஸங்க்யானம் உபயுஜ்யதே

காமதந்த்ரே அர்த்ததந்த்ரே ச காந்தர்வே நாடகேபி வா

ஸூபஷாஸ்த்ரே ததா வைத்யே வாஸ்துவித்யாதிவஸ்துஷு

சந்த அலங்கார காவ்யேஷு தர்க்க வ்யாகரணாதிஷு

கலாகுணேஷு ஸர்வேஷு ப்ரஸ்துதம் கணிதம் பரம்

லௌகிகத்தில் வேத கல்வியில் ஸமயத்தில்

வியாபாரத்தில் எதிலும் கணிதம் உபயோகப்படும்

காமதந்திரத்தில் அர்த்ததந்திரத்தில் காந்தர்வத்தில் (இசையில்) நாடகத்தில் (கூத்தில்)

சமையலில் (ஸூபசாத்திரம்) மருத்துவத்தில் (வைத்தியம்) கட்டடகலையில் (வாஸ்து)

சந்ததில் அலங்காரத்தில் காவியத்தில் தர்க்கத்தில் இலக்கணத்தில் (வ்யாகர்ணம்)

கலைகுணங்களில் எங்கிலும் கணிதம் பரம்புகழ் என்பர்.