Astronomy and Mathematics in Ancient India

Astronomy


* Earliest known precise celestial calculations:

As argued by James Q. Jacobs, Aryabhata, an Indian Mathematician (c. 500AD) accurately calculated celestial constants like earth's rotation per solar orbit, days per solar orbit, days per lunar orbit. In fact, to the best of my knowledge, no source from prior to the 18th century had more accurate results on the values of these constants! Click here for details. Aryabhata's 499 AD computation of pi as 3.1416 (real value 3.1415926...) and the length of a solar year as 365.358 days were also extremely accurate by the standards of the next thousand years.

* Astronomical time spans:
The notion of of time spans that are truly gigantic by modern standards are rarely found in ancient civilizations as the notion of large number is rare commodity. Apart from the peoples of the Mayan civilization, the ancient Hindus appear to be the only people who even thought beyond a few thousand years. In the famed book Cosmos, physicist-astronomer-teacher Carl Sagan writes "... The dates on Mayan inscriptions also range deep into the past and occasionally far into the future. One inscription refers to a time more than a million years ago and another perhaps refers to events of 400 million years ago, ... The events memorialized may be mythical, but the time scales are pridigious". Hindu scriptures refer to time scales that vary from ordinary earth day and night to the day and night of the Brahma that are a few billion earth years long. Sagan continues, "A millennium before Europeans were wiling to divest themselves of the Biblical idea that the world was a few thousand years old, the Mayans were thinking of millions and the Hindus billions" [See 5].

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

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

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

Hinduism is the only religion that propounds the idea of life-cycles of the universe. It suggests that the universe undergoes an infinite number of deaths and rebirths. Hinduism, according to Sagan, "... is the only religion in which the time scales correspond... to those of modern scientific cosmology. Its cycles run from our ordinary day and 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 time since the Big Bang" [See 5]. Long before Aryabhata (6th century) came up with this awesome achievement, apparently there was a mythological angle to this as well -- it becomes clear when one looks at the following translation of Bhagavad Gita (part VIII, lines 16 and 17), "All the planets of the universe, from the most evolved to the most base, are places of suffering, where birth and death takes place. But for the soul that reaches my Kingdom, O son of Kunti, there is no more reincarnation. One day of Brahma is worth a thousand 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 of creation. In other words, the four ages of the mahayuga must be repeated a thousand times to make a "day ot Brahma", a unit of time that is the equivalent of 4.32 billion human years, doubling which one gets 8.64 billion years for a Brahma day and night. This was later theorized (possibly independently) by Aryabhata in the 6th century. The cyclic nature of this analysis suggests a universe that is expanding 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 the Big Bang happened and it is believed that the universe, as we "know" it, came into existence; one that is continually expanding after the Big Bang. That the galaxies are receding from us can be proved by showing Dopler shifts of far off galaxies. Common belief is that it happened from a mathematical point with no dimension at all. All the matter in our universe was concentrated in that miniscule volume. Although we know that we are living in an expanding universe, physicists are not sure whether it will always be expanding. This is because it is not known whether there is enough matter in the universe such that there is enough gravitational cohesion in it that the expansion will gradually slow down, stop and reverse itself resulting into a contracting universe. If we live in such an oscillating universe, then the Big Bang is not the beginning or creation of the universe, but merely the end of the previous cycle, the destruction of the last incarnation of the universe in the very way suggested by Hindu philosophers thousands of years ago!

A brand new theory -- that of a "CYCLIC MODEL", developed by Princeton University's Paul Steinhardt and Cambridge University's Neil Turok, made its highest-profile appearance yet in April 2002, on Science Express, the Web site for the journal Science. But past incarnations of the idea have been hotly debated within the cosmological community from 2001. A jist of the claims can be found here. The PDF preprint of the entire paper can be downloaded from here. The Hindu belief that the Universe has no beginning or end, but follows a cosmic creation and dissolution can be found here.

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

Mathematics/Computer Science

* Binary System of number representation:
A Mathematician named Pingala (c. 100BC) developed a system of binary enumeration convertible to decimal numerals [See 3]. He described the system in his book called Chandahshaastra. The system he described is quite similar to that of Leibnitz, who was born in the 17th century.

* Earliest and only known Modern Language:
Panini (c 400BC), in his Astadhyayi, gave formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 fundamental elements, like nouns, verbs, vowels and consonents, he put them into classes. The construction of sentences, compound nouns etc. was explained as ordered rules operating on underlying fundamental structures. This is exactly in congruence 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 as aphorisms), he built virtually the whole structure of the Sanskrit language. He used a notation precisely as powerful as the Backus normal form, an algabraic notation used in Computer Science to represent numerical and other patterns by letters.

It is my contention that because of the scientific nature of the method of pronunciation of the vowels and consonents in the Indian languages (specially those coming directly from Pali, Prakit and Sanskrit), every part of the mouth is exercised during speaking. This results into speakers of Indian languages being able to pronounce words from any language. This is unlike the case with say native English speakers, as their tongue becomes unused to being able to touch certain portions of the mouth during pronunciation, thus giving the speakers a hard time to speak certain words from a language 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 is one.

* Invention of Zero:
Although ancient Babylonians were known to have used what is often called "place holders" to distinguish between numbers like 809 and 89, they were nothing more than blank spaces or at times two wedge shapes like ". More importantly, they lacked the realization that zero has a place in the number system as well as it comes with a baggage of abstract interpretations. Hence, while they can be credited with intelligently solving a practical problem of avoiding misinterpreting two numbers like 809 and 89, they can hardly be credited with the invention of the complex notion of zero and the even more complex notion of the abstract idea of "nothingness".

The ancient Greeks were beginning their contributions to mathematics around the time zero as an empty place holder was being popularized by Babylonian mathematicians. The Greeks did not adopt what is called a positional number system, a system that gave a value to a number because of its relative position in the set of numerals. This is because the Greeks' achievements were based on geometry. This resulted into firstly, Greeks relating numbers with lengths of line segments, and secondly, decoupling numbers from any potential abstract interpretations. It is commonly thought that in Greek society numbers that required to be "named" were not used by mathematician- philosophers, but by merchants and hence no clever notation was needed. Thus even the eminent mathematician like Ptolemy used the then recent place holder "zero" more as a punctuation mark than any serious numeral. Although a few Greek astronomers began using the symbol "O", the symbol more familiar to us now, to denote place holders, zero was not thought of as a number by the Greeks.

The first notions of zero as a number and its uses have been found in ancient Mathematical treatise from India and thus India is correctly related to the immensely important mathematical discovery of the numeral zero. This concept, combined with the place-value system of enumeration, became the basis for a classical era renaissance in Indian mathematics. Indians began using zero both as a number in the place-value system of numerals as well as to denote an empty place (place holder). Obviously, the use as a number came later. Aryabhata devised a number system what has no zero yet a positional number system. There is however, evidence that z dot has been used in earlier Indian manuscripts to denote an empty position. Also contemporary Indian scriptures also tend to use zero in places where unknown values are registered, where we would use x. Later Indian mathematicians had names for zero, but no symbol for it. Aryabhata used the word "kha" for position and it was also used later as the name for zero.

The oldest known text to use zero is an Indian (Jaina) text entitled the Lokavibhaaga ("The Parts of the Universe"), which has been definitely 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 great leap from physical to abstract that one needs to bridge when dealing with zero. With zero also comes the notion of negative numbers and along with all these comes a series of related questions about arithmetic operations on natural numbers, both positive and negative and zero.

The development of the notion of zero began, in my opinion, when mathematicians tried to answer these questions. Three Hindu mathematicians, Brahmagupta, Mahavira and Bhaskara tried to answer these in their treatise. In the 7th century Brahmagupta attempted to provide rules for addition and subtraction involving zero.

The sum of zero and a negative number is negative, the sume of a positive number and zero if positive, the sum of zero and zero is zero. A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is nagative, zero subtracted from a positive number 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 wrongly gives zero divided by zero to be zero. Brahmagupta's is the first attempt from any mathematician to explain the arithmetic operations on natural numbers and zero.

In the 9th century, Mahavira updated Brahmagupta's attempts at defining operations using zero. Although he correctly finds out that a number multiplied by zero is zero, but wrongly says that a number remains 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 of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.

This, in its face value seems correct, by suggesting that any number when divided by zero is infinity, Bhaskara suggeted that zero multiplied by infinity is any number, and hence all numbers are equal, which is not correct. But Bhaskara did correctly find out that the square of zero is zero, as is the square root.

The Indian numeral system and its place value, decimal system of enumeration came to the attention of the Arabs in the seventh or eighth century, and served as the basis for the well known advancement in Arab mathematics, represented by figures such as al-Khwarizmi. Al-Khwarizmi wrote Al'Khwarizmi on the Hindu Art of Reckoning that described the Indian place-value system of numerals based 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 the 12th century al- Samawal extended arithmetic operations using zero as follows.

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

Zero also reached eastwards from India to China, where Chinese scholars Chin Chiu-Shao and Chu Shih-Chieh made use of the same symbol O for a places-based system in the 12th and 13th centuries respectively. From the time of Han (206 to 220 BC), Chinese scholars used a place-value system called the suan zi ("calculation using rods") that was a regular system that used horizontal and vertical lines that used to denote the nine numerals. Ifrah says that "Thus one could be forgiven for assuming that following the links established between India and China at the beginning of the beginning of the first millennium BC, Indian scholars were influenced by Chinese mathematicians to create their own system in an imitation of the Chinese counting method." [See 4] He goes on to argue 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 Indian mathematicians vis-a-vis their Chinese counterparts. Ifra also establishes that the Chinese scholars overcame the difficulties the absence of zeros caused in trying to represent numbers like 1,270,000 often either using characters of their ordinary counting system (a non-positional system that did not require the use of a zero) or simply by empty spaces. After providing a sequence of clues, [in 4], Ifrah continues "It was only after the eighth century BC, and doubtless due to the influence of the Indian Buddhist missionaries, that Chinese mathematicians introduced the use of zero in the form of a little circle or dot (signs that originated in India),...".

Zero reached Europe in the twelfth century when Adelard of Bath translated al-Khwarizmi's works into Latin [See 1]. Fibonacci was one of the main mathematicians who accepted the concepts of zero in Europe. He was an important link between the Hindu-Arabic number system. In his treatise Liber Abaci ("a tract about the abacus"), published in 1202, he described the nine Indian symbols together with the symbol O for zero, but it was not widely accepted until much later. Significantly, Fibonacci spoke of numbers 1 through 9, but a "sign" O. Although he brought the notion of zero to Europe, it is clear that he was not able to reach the sophistication of Indians like Brahamagupta, Mahavira and Bhaskara, nor of the Arabic mathematicians like al-Samawal. The Europeans were at first resistant to this system, being attached to the far less logical Roman numeral system (notably the Romans never propounded the idea of zero), but their eventual adoption of this system arguably led to the scientific revolution that began to sweep Europe beginning by the middle of the second millennium. However, it was not until the 17th century that zero found widespread acceptance through a lot of resistance.

* The word "Algorithm":

Al-Khwarizmi, an eminent 9th century Arab scholar, played important roles in importing knowledge on arithematic and algebra from India to the Arabs. In his work, De numero indorum (Concerning the Hindu Art of Reckoning), it was based presumably on an Arabic translation of Brahmagupta where he gave a full account of the Hindu numerals which was the first to expound the system with its digits 0,1,2,3,...,9 and decimal place value which was a fairly recent arrival from India. Because of this book with the Latin translations made a false inquiry that our system of numeration is arabic in origin. The new notation came to be known as that of al-Khwarizmi, or more carelessly, algorismi; ultimately the scheme of numeration making use of the Hindu numerals came to be called simply algorism or algorithm, a word that, originally derived from the name al-Khwarizmi, now means, more generally, any peculiar rule of procedure or operation. The Hindu numerals like much new mathematics were not welcomed by all. Click here for details.

* Representing Large numbers:

Mathematicians in India invented the base ten system in ancient times. But research did not stop there. The practice of representing large numbers also evolved in ancient India. The base ten system of calculation that uses nine numerals and the zero stood as an efficient way to represent numbers ranging from a very small decimal to an inconceivably large number. The biggest number known to Greeks was the myriad (10,000) whereas the Chinese, until recent times, had 10,000 as the largest unit of enumeration and the ancient Arabs knew only until 1,000. The notion of representing large numbers as powers of 10, one that was invented in India, turned out to be extremely handy. The Yajur Veda Samhitaa, one of the Vedic texts written at least 1,000 years before Euclid 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 their Greek contmporaries, mainly because of the latter's handicap of not being able to accept the fundamental Mathematical notion of abstract numerals. The place value system is built into the Sanskrit language and so whereas in English we only use thousand, million, billion etc, in Sanskrit there are specific nomenclature for the powers 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 such a practice were two-folds. Firstly, the removal of special imporatance of numbers. Instead of naming numbers in grops of three, four or eight orders of units one could use the necessary name for the power of 10. Secondly, the notion of the term "of the order of". To express the order of a particular number, one simply needs to use the nearest two powers of 10 to express its enormity.

Evidences of using very large numbers have been found in the Vedas which are ancient Hindu scriptures. Vedas are the most ancient written texts written in any Indo-European language. They were written in Sanskrit from around 500BC, although traces go back to 2000BC [See 4]. In the Taittiriya Upanishad, which is a part of the third Veda, Yajur Veda, there is a section (anuvaka), that extols the "Beatific Calculus" or a quasi-mathematical relationship between bliss of a young man, who has everything in the world to the bliss of the Brahman, or "realization". Translated roughly as follows, summarized from one done by Max Muller, firstly it says that fear is all-pervasive. It continues by assuming that a young, good man who is fit, healthy and strong, and has all the wealth in the world, is one unit of human bliss. The anuvaka provides a precise calculation of a series of multiplications by 100 to give number 10010 units of human bliss that can be had when one attains Brahman. The previous anuvaka exhorts the aspirants to be fearless and strong, as only such a person may realize the absolute within.



* "... true birthplace of our numerals": Georges Ifrah:

Famed French scholar Georges Ifrah spent years travelling and studying the mystery of the evolution of numbers. While it is hard to prove that India is truly the birthplace of our modern numerals, in my brief survey of the topic, it seems that there is no better authority in the field other than Ifrah. I would refer the interested reader to his authoritative book [See 4] to get a crisp, yet convincing account supporting his claims. Ifrah provides a total of 45 pieces of evidences, supported by numerous research work from contemporary scholars. Of the 45, 17 are from scholarly work from Europe that includes work of scholars like Laplace, Fibonacci, and Adelard of Bath, and 28 are from work from Arabic sources that includes work of scholars like al Biruni. He refers to 24 evidences from scriptures from India, whose dates range from 1150 BC until 458 BC, when the Jaina text Lokavibhaaga dates back to. Of particular interest was the work by Bhaskaracharya (1150 BC) where he makes a reference to zero and the Indian place-value system as being creations of Brahma, indicating that by that time they were considered "to have always been used by humans, and thus to have constituted a "revelation" of the divinities", [See 4]. Ifrah goes on to explain, with furious objectivity aided by a plethora of evidences that are not isolated pieces of information, but "a huge collection of proofs from all disciplines, dating from the most significant eras", to establish his claim. He also shows how the numerals evolved to look as they look today. His suggested pathway to the modern numerals is:



* 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 inventors of this fundamental discovery, which is no less important than such feats as the mastery of fire, the development of agriculture, or the invention of the wheel, writing or the steam engine, were the mathematicians and astronomers of the Indian civilisation: scholars who, unlike the Greeks, were concerned with practical applications and who were motivated by a kind of passion for both numbers and numerical calculations."





References



[1] B. V. Subbarayappa. "India's Contributions to the History of Science" in Lokesh Chandra, et al., eds. India's Contribution

to World Thought and Culture. Madras: Vivekananda Rock Memorial Committee, pp47-66, 1970.



[2] G. G. Joseph. The crest of the peacock: Non-European Roots of Mathematics. 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 by David Bellos, E. F. Harding, Sophie Wood and Ian Monk. The Harvill Press, London, 1998.



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



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



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



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