Astronomy and Mathematics in Ancient India

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Astronomy and Mathematics inAncient IndiaURL: http://www.cerc.utexas.edu/~jay/india_science.html

Astronomy

* Earliest known precise celestial calculations:As argued by James Q. Jacobs, Aryabhata, an Indian Mathematician(c. 500AD) accurately calculated celestial constants like earth'srotation per solar orbit, days per solar orbit, days per lunar orbit. Infact, to the best of my knowledge, no source from prior to the 18thcentury had more accurate results on the values of these constants!

Click here for details. Aryabhata's 499 AD computation of pi as3.1416 (real value 3.1415926...) and the length of a solar year as365.358 days were also extremely accurate by the standards of thenext thousand years.

* Astronomical time spans:The notion of of time spans that are truly gigantic by modernstandards are rarely found in ancient civilizations as the notion oflarge number is rare commodity. Apart from the peoples of theMayan civilization, the ancient Hindus appear to be the only people

who even thought beyond a few thousand years. In the famed bookCosmos, physicist-astronomer-teacher Carl Sagan writes "... Thedates on Mayan inscriptions also range deep into the past andoccasionally far into the future. One inscription refers to a time morethan a million years ago and another perhaps refers to events of400 million years ago, ... The events memorialized may be mythical,but the time scales are pridigious". Hindu scriptures refer to timescales that vary from ordinary earth day and night to the day andnight of the Brahma that are a few billion earth years long. Sagancontinues, "A millennium before Europeans were wiling to divest

themselves of the Biblical idea that the world was a few thousandyears old, the Mayans were thinking of millions and the Hindusbillions" [See 5].

* Theory of creation of the universe:A 9th century Hindu scripture, The Mahapurana by Jinasena claimsthe something as modern as the following: (translation from [5])


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Some foolish men declare that a Creator made the world. Thedoctrine that the world was created is ill-advised, and should berejected. If God created the world, where was he before creation?...How could God have made the world without any raw material? Ifyou say He made this first, and then the world, you are faced withan endless regression... Know that the world is uncreated, as timeitself is, without beginning and end. And it is based on principles.

Theories of the creation of universe are present in almost everyculture. Mostly they represent some story portraying creation frommating of Gods or humans, or from some divine egg, essentially allof them reflecting the human endeavour to provide explanations toa grave scientific question using common human experience.

Hinduism is the only religion that propounds the idea of life-cycles ofthe universe. It suggests that the universe undergoes an infinite

number of deaths and rebirths. Hinduism, according to Sagan, "... isthe only religion in which the time scales correspond... to those ofmodern scientific cosmology. Its cycles run from our ordinary dayand night to a day and night of the Brahma, 8.64 billion years long,longer than the age of the Earth or the Sun and about half the timesince the Big Bang" [See 5]. Long before Aryabhata (6th century)came up with this awesome achievement, apparently there was amythological angle to this as well -- it becomes clear when one looksat the following translation of Bhagavad Gita (part VIII, lines 16 and

17), "All the planets of the universe, from the most evolved to themost base, are places of suffering, where birth and death takesplace. But for the soul that reaches my Kingdom, O son of Kunti,there is no more reincarnation. One day of Brahma is worth athousand of the ages [yuga] known to humankind; as is each night."

Thus each kalpa is worth one day in the life of Brahma, the God ofcreation. In other words, the four ages of the mahayuga must berepeated a thousand times to make a "day ot Brahma", a unit oftime that is the equivalent of 4.32 billion human years, doublingwhich one gets 8.64 billion years for a Brahma day and night. This

was later theorized (possibly independently) by Aryabhata in the 6thcentury. The cyclic nature of this analysis suggests a universe that isexpanding to be followed by contraction... a cosmos without end.

This, according to modern physicists is not an impossibility.

And here is how -- a few billion years ago, something known as theBig Bang happened and it is believed that the universe, as we"know" it, came into existence; one that is continually expanding


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after the Big Bang. That the galaxies are receding from us can beproved by showing Dopler shifts of far off galaxies. Common belief isthat it happened from a mathematical point with no dimension atall. All the matter in our universe was concentrated in that minisculevolume. Although we know that we are living in an expandinguniverse, physicists are not sure whether it will always beexpanding. This is because it is not known whether there is enoughmatter in the universe such that there is enough gravitationalcohesion in it that the expansion will gradually slow down, stop andreverse itself resulting into a contracting universe. If we live in suchan oscillating universe, then the Big Bang is not the beginning orcreation of the universe, but merely the end of the previous cycle,the destruction of the last incarnation of the universe in the veryway suggested by Hindu philosophers thousands of years ago!

A brand new theory -- that of a "CYCLIC MODEL", developed byPrinceton University's Paul Steinhardt and Cambridge University'sNeil Turok, made its highest-profile appearance yet in April 2002, onScience Express, the Web site for the journal Science. But pastincarnations of the idea have been hotly debated within thecosmological community from 2001. A jist of the claims can befound here. The PDF preprint of the entire paper can be downloadedfrom here. The Hindu belief that the Universe has no beginning orend, but follows a cosmic creation and dissolution can be foundhere.

* Earth goes round the sun:Aryabhata, it so happens, was apparently quite sceptical of thewidely held doctrines about eclipses and also about the belief thatthe Sun goes round the Earth. He didn't think that eclipses werecaused by Rahu but by the Earth's shadow over the Moon and theMoon obscuring the Sun. As early as the sixth century, he talked ofthe diurnal motion of the earth and the appearance of the Sun goinground it.

Mathematics/Computer Science* Binary System of number representation:A Mathematician named Pingala (c. 100BC) developed a system ofbinary enumeration convertible to decimal numerals [See 3]. Hedescribed the system in his book called Chandahshaastra. Thesystem he described is quite similar to that of Leibnitz, who wasborn in the 17th century.


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* Earliest and only known Modern Language:Panini (c 400BC), in his Astadhyayi, gave formal production rulesand definitions to describe Sanskrit grammar. Starting with about1700 fundamental elements, like nouns, verbs, vowels andconsonents, he put them into classes. The construction ofsentences, compound nouns etc. was explained as ordered rulesoperating on underlying fundamental structures. This is exactly incongruence with the fundamental notion of using terminals, non-terminals and production rules of moderm day Computer Science.On the basis of just under 4,000 sutras (rules expressed asaphorisms), he built virtually the whole structure of the Sanskritlanguage. He used a notation precisely as powerful as the Backusnormal form, an algabraic notation used in Computer Science torepresent numerical and other patterns by letters.

It is my contention that because of the scientific nature of themethod of pronunciation of the vowels and consonents in the Indianlanguages (specially those coming directly from Pali, Prakit andSanskrit), every part of the mouth is exercised during speaking. Thisresults into speakers of Indian languages being able to pronouncewords from any language. This is unlike the case with say nativeEnglish speakers, as their tongue becomes unused to being able totouch certain portions of the mouth during pronunciation, thusgiving the speakers a hard time to speak certain words from alanguage not sharing a common ancestry with English. I am not

aware of any theory in these lines, but I would like to know if there isone.

* Invention of Zero:Although ancient Babylonians were known to have used what isoften called "place holders" to distinguish between numbers like 809and 89, they were nothing more than blank spaces or at times twowedge shapes like ". More importantly, they lacked the realizationthat zero has a place in the number system as well as it comes witha baggage of abstract interpretations. Hence, while they can be

credited with intelligently solving a practical problem of avoidingmisinterpreting two numbers like 809 and 89, they can hardly becredited with the invention of the complex notion of zero and theeven more complex notion of the abstract idea of "nothingness".

The ancient Greeks were beginning their contributions tomathematics around the time zero as an empty place holder wasbeing popularized by Babylonian mathematicians. The Greeks did


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not adopt what is called a positional number system, a system thatgave a value to a number because of its relative position in the setof numerals. This is because the Greeks' achievements were basedon geometry. This resulted into firstly, Greeks relating numbers withlengths of line segments, and secondly, decoupling numbers fromany potential abstract interpretations. It is commonly thought that inGreek society numbers that required to be "named" were not usedby mathematician- philosophers, but by merchants and hence noclever notation was needed. Thus even the eminent mathematicianlike Ptolemy used the then recent place holder "zero" more as apunctuation mark than any serious numeral. Although a few Greekastronomers began using the symbol "O", the symbol more familiarto us now, to denote place holders, zero was not thought of as anumber by the Greeks.

The first notions of zero as a number and its uses have been foundin ancient Mathematical treatise from India and thus India iscorrectly related to the immensely important mathematicaldiscovery of the numeral zero. This concept, combined with theplace-value system of enumeration, became the basis for a classicalera renaissance in Indian mathematics. Indians began using zeroboth as a number in the place-value system of numerals as well asto denote an empty place (place holder). Obviously, the use as anumber came later. Aryabhata devised a number system what hasno zero yet a positional number system. There is however, evidence

that z dot has been used in earlier Indian manuscripts to denote anempty position. Also contemporary Indian scriptures also tend to usezero in places where unknown values are registered, where wewould use x. Later Indian mathematicians had names for zero, butno symbol for it. Aryabhata used the word "kha" for position and itwas also used later as the name for zero.

The oldest known text to use zero is an Indian (Jaina) text entitledthe Lokavibhaaga ("The Parts of the Universe"), which has beendefinitely dated to 25 August 458 BC [See 4] An inscription, created

in 876AD, found in Gwalior, acts as the first use of zero as a number.Zero is not a "natural" candidate for being a number. It is a greatleap from physical to abstract that one needs to bridge whendealing with zero. With zero also comes the notion of negativenumbers and along with all these comes a series of relatedquestions about arithmetic operations on natural numbers, bothpositive and negative and zero.


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The development of the notion of zero began, in my opinion, whenmathematicians tried to answer these questions. Three Hindumathematicians, Brahmagupta, Mahavira and Bhaskara tried toanswer these in their treatise. In the 7th century Brahmaguptaattempted to provide rules for addition and subtraction involvingzero.

The sum of zero and a negative number is negative, the sume of apositive number and zero if positive, the sum of zero and zero iszero. A negative number subtracted from zero is positive, a positivenumber subtracted from zero is negative, zero subtracted from anegative number is nagative, zero subtracted from a positivenumber is positive, zero subtracted from zero is zero.

Brahmagupta also says that any number multiplied by zero is zero.But problems arise when he tries to explain division. While he is

unsure about what division of a number by zero means, he wronglygives zero divided by zero to be zero. Brahmagupta's is the firstattempt from any mathematician to explain the arithmeticoperations on natural numbers and zero.

In the 9th century, Mahavira updated Brahmagupta's attempts atdefining operations using zero. Although he correctly finds out that anumber multiplied by zero is zero, but wrongly says that a numberremains unchanged when divided by zero.

The next valiant attempt came from Bhaskara in the 11th century.

Division of zero still remained an illusive mystery.

A quantity divided by zero becomes a fraction the denominator ofwhich is zero. This fraction is termed an infinite quantity. In thisquantity consisting of that which has zero for its divisor, there is noalteration, though many may be inserted or extracted; as no changetakes place in the infinite and immutable God when worlds arecreated or destroyed, though numerous orders of beings areabsorbed or put forth.

This, in its face value seems correct, by suggesting that any number

when divided by zero is infinity, Bhaskara suggeted that zeromultiplied by infinity is any number, and hence all numbers areequal, which is not correct. But Bhaskara did correctly find out thatthe square of zero is zero, as is the square root.

The Indian numeral system and its place value, decimal system ofenumeration came to the attention of the Arabs in the seventh oreighth century, and served as the basis for the well known


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advancement in Arab mathematics, represented by figures such asal-Khwarizmi. Al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art ofReckoning that described the Indian place-value system of numeralsbased on numerals 1 through 9 and 0. Scholars like ibn Ezra and al-Samawal used the notion of zero from al-Khwarizmi's work. In the12th century al- Samawal extended arithmetic operations using zeroas follows.

If we subtract a positive number from zero the same negativenumber remains, ... if we subtract a negative number from zero thesame positive number remains.

Zero also reached eastwards from India to China, where Chinesescholars Chin Chiu-Shao and Chu Shih-Chieh made use of the samesymbol O for a places-based system in the 12th and 13th centuriesrespectively. From the time of Han (206 to 220 BC), Chinese

scholars used a place-value system called the suan zi ("calculationusing rods") that was a regular system that used horizontal andvertical lines that used to denote the nine numerals. Ifrah says that"Thus one could be forgiven for assuming that following the linksestablished between India and China at the beginning of thebeginning of the first millennium BC, Indian scholars were influencedby Chinese mathematicians to create their own system in animitation of the Chinese counting method." [See 4] He goes on toargue that in suan zi, the zero appeared at a much later date. Thus

the notion of zero helps one to recognize the originality of the Indianmathematicians vis-a-vis their Chinese counterparts. Ifra alsoestablishes that the Chinese scholars overcame the difficulties theabsence of zeros caused in trying to represent numbers like1,270,000 often either using characters of their ordinary countingsystem (a non-positional system that did not require the use of azero) or simply by empty spaces. After providing a sequence ofclues, [in 4], Ifrah continues "It was only after the eighth century BC,and doubtless due to the influence of the Indian Buddhistmissionaries, that Chinese mathematicians introduced the use of

zero in the form of a little circle or dot (signs that originated inIndia),...".

Zero reached Europe in the twelfth century when Adelard of Bathtranslated al-Khwarizmi's works into Latin [See 1]. Fibonacci was oneof the main mathematicians who accepted the concepts of zero inEurope. He was an important link between the Hindu-Arabic numbersystem. In his treatise Liber Abaci ("a tract about the abacus"),


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published in 1202, he described the nine Indian symbols togetherwith the symbol O for zero, but it was not widely accepted untilmuch later. Significantly, Fibonacci spoke of numbers 1 through 9,but a "sign" O. Although he brought the notion of zero to Europe, itis clear that he was not able to reach the sophistication of Indianslike Brahamagupta, Mahavira and Bhaskara, nor of the Arabicmathematicians like al-Samawal. The Europeans were at firstresistant to this system, being attached to the far less logical Romannumeral system (notably the Romans never propounded the idea ofzero), but their eventual adoption of this system arguably led to thescientific revolution that began to sweep Europe beginning by themiddle of the second millennium. However, it was not until the 17thcentury that zero found widespread acceptance through a lot ofresistance.

* The word "Algorithm":Al-Khwarizmi, an eminent 9th century Arab scholar, playedimportant roles in importing knowledge on arithematic and algebrafrom India to the Arabs. In his work, De numero indorum(Concerning the Hindu Art of Reckoning), it was based presumablyon an Arabic translation of Brahmagupta where he gave a fullaccount of the Hindu numerals which was the first to expound thesystem with its digits 0,1,2,3,...,9 and decimal place value whichwas a fairly recent arrival from India. Because of this book with theLatin translations made a false inquiry that our system of

numeration is arabic in origin. The new notation came to be knownas that of al-Khwarizmi, or more carelessly, algorismi; ultimately thescheme of numeration making use of the Hindu numerals came tobe called simply algorism or algorithm, a word that, originallyderived from the name al-Khwarizmi, now means, more generally,any peculiar rule of procedure or operation. The Hindu numerals likemuch new mathematics were not welcomed by all. Click here fordetails.

* Representing Large numbers:

Mathematicians in India invented the base ten system in ancienttimes. But research did not stop there. The practice of representinglarge numbers also evolved in ancient India. The base ten system ofcalculation that uses nine numerals and the zero stood as anefficient way to represent numbers ranging from a very smalldecimal to an inconceivably large number. The biggest numberknown to Greeks was the myriad (10,000) whereas the Chinese,


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until recent times, had 10,000 as the largest unit of enumerationand the ancient Arabs knew only until 1,000. The notion ofrepresenting large numbers as powers of 10, one that was inventedin India, turned out to be extremely handy. The Yajur VedaSamhitaa, one of the Vedic texts written at least 1,000 years beforeEuclid lists names for each of the units of ten upto the twelfth power[See 1]. Later other Indian texts (from Buddhist and Jaina authors)extended this list as high as the 53rd power, far exceeding theirGreek contmporaries, mainly because of the latter's handicap of notbeing able to accept the fundamental Mathematical notion ofabstract numerals. The place value system is built into the Sanskritlanguage and so whereas in English we only use thousand, million,billion etc, in Sanskrit there are specific nomenclature for thepowers of 10, most used in modern times are dasa (10), sata (100),

sahasra (1,000=1K), ayuta (10K), laksha (100K), niyuta (106=1M),koti (10M), vyarbuda (100M), paraardha (1012) etc. Results of sucha practice were two-folds. Firstly, the removal of specialimporatance of numbers. Instead of naming numbers in grops ofthree, four or eight orders of units one could use the necessaryname for the power of 10. Secondly, the notion of the term "of theorder of". To express the order of a particular number, one simplyneeds to use the nearest two powers of 10 to express its enormity.

Evidences of using very large numbers have been found in theVedas which are ancient Hindu scriptures. Vedas are the most

ancient written texts written in any Indo-European language. Theywere written in Sanskrit from around 500BC, although traces goback to 2000BC [See 4]. In the Taittiriya Upanishad, which is a partof the third Veda, Yajur Veda, there is a section (anuvaka), thatextols the "Beatific Calculus" or a quasi-mathematical relationshipbetween bliss of a young man, who has everything in the world tothe bliss of the Brahman, or "realization". Translated roughly asfollows, summarized from one done by Max Muller, firstly it says thatfear is all-pervasive. It continues by assuming that a young, goodman who is fit, healthy and strong, and has all the wealth in theworld, is one unit of human bliss. The anuvaka provides a precisecalculation of a series of multiplications by 100 to give number10010 units of human bliss that can be had when one attainsBrahman. The previous anuvaka exhorts the aspirants to be fearlessand strong, as only such a person may realize the absolute within.


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* "... true birthplace of our numerals": Georges Ifrah:Famed French scholar Georges Ifrah spent years travelling andstudying the mystery of the evolution of numbers. While it is hard toprove that India is truly the birthplace of our modern numerals, inmy brief survey of the topic, it seems that there is no betterauthority in the field other than Ifrah. I would refer the interestedreader to his authoritative book [See 4] to get a crisp, yetconvincing account supporting his claims. Ifrah provides a total of 45pieces of evidences, supported by numerous research work fromcontemporary scholars. Of the 45, 17 are from scholarly work fromEurope that includes work of scholars like Laplace, Fibonacci, andAdelard of Bath, and 28 are from work from Arabic sources thatincludes work of scholars like al Biruni. He refers to 24 evidencesfrom scriptures from India, whose dates range from 1150 BC until

458 BC, when the Jaina text Lokavibhaaga dates back to. Ofparticular interest was the work by Bhaskaracharya (1150 BC) wherehe makes a reference to zero and the Indian place-value system asbeing creations of Brahma, indicating that by that time they wereconsidered "to have always been used by humans, and thus to haveconstituted a "revelation" of the divinities", [See 4]. Ifrah goes on toexplain, with furious objectivity aided by a plethora of evidencesthat are not isolated pieces of information, but "a huge collection ofproofs from all disciplines, dating from the most significant eras", toestablish his claim. He also shows how the numerals evolved to look

as they look today. His suggested pathway to the modern numeralsis:

* Brahmi (often called the "mother" of all Indian writing) numerals* Shaka, Kushana inscriptions* Gupta style* Nagari style* Arabic from the "Gubar" style* European late middle ages (cursive forms of the Algorisms)* modern.

Ifrah salutes the Indian researchers saying that the "...real inventorsof this fundamental discovery, which is no less important than suchfeats as the mastery of fire, the development of agriculture, or theinvention of the wheel, writing or the steam engine, were themathematicians and astronomers of the Indian civilisation: scholarswho, unlike the Greeks, were concerned with practical applicationsand who were motivated by a kind of passion for both numbers and


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numerical calculations."

References

[1] B. V. Subbarayappa. "India's Contributions to the History of

Science" in Lokesh Chandra, et al., eds. India's Contributionto World Thought and Culture. Madras: Vivekananda Rock MemorialCommittee, pp47-66, 1970.

[2] G. G. Joseph. The crest of the peacock: Non-European Roots ofMathematics. Princeton University Press, 1991.

[3] B. van Nooten. "Binary Numbers in Indian Antiquity" in T. R. N.Rao and Subhash Kak, editors. Computing Science in Ancient India,pp. 21-39.

[4] G. Ifrah. The Universal History of Numbers: From Prehistory to

the Invention of the Computer. Translated from French to English byDavid Bellos, E. F. Harding, Sophie Wood and Ian Monk. The HarvillPress, London, 1998.

[5] C. Sagan. Cosmos. Ballantine Books, New York, 1980.

[6] R. Kaplan. The Nothing that Is: A Natural History of Zero. OxfordUniversity Press, 2000. Bibliography and notes: click here.

[7] C. Seife. Zero: The Biography of a Dangerous Idea. Viking, 2000.

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Astronomy and Mathematics in Ancient India