Bhaskara I.

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Bhaskara , also Bhaskara I. , (* around 600 in Saurashtra  ?, Gujarat , † around 680 in Ashmaka ) was an Indian mathematician and astronomer .


Almost nothing is known about Bhaskara's life. He received his astronomical training from his father. Bhaskara is considered to be the most important representative of the astronomical school founded by Aryabhata .

Representation of numbers

Perhaps Bhaskara's most important mathematical achievement concerns the representation of numbers in place value systems . Indian astronomers are already familiar with the first representations of significance around 500 . The numbers are not yet written in digits, but in word numbers or symbols and held in verse. For example, the moon is specified for the number 1, since it only exists once; wings, twins or eyes apply to the number 2, since they always appear as a pair; the number 5 stands for the (five) senses (cf. today's mnemonics ). These words are strung together in a similar way to our current decimal system , only they started with the ones and ended with the highest power of ten, contrary to the original Brahmi spelling. For example is

1052 = wing meaning void moon.

Why did the Indian scientists use numerals and not the Brahmi numbers that were also known at the time ? The texts were written in Sanskrit , the "language of the gods", which in India played a role similar to that of Latin today in Europe, and completely different dialects were spoken. It is likely that the Brahmi numerals used in everyday life were initially felt to be too vulgar for the gods (Ifrah 2000, p. 431).

Aryabhata later used a different method (" Aryabhata code ") around 510 , in which he designated the numbers with syllables. His system is based on the base 100, and not 10 (Ifrah 2000, p. 449). In his commentary on Aryabhatiya from the year 629, Bhaskara modified Aryabhata's syllable numeric notation to a place value system based on base 10, which contains a zero . However, he uses fixed numerals, starting with the ones, then with the tens, etc., for example he writes the number 4320000 as

viyat ambara akasha sunya yama rama veda
sky the atmosphere ether Empty Primeval couple ( Yama and Yami ) Rama Veda
0 0 0 0 2 3 4th

His system is really positional, as the same words that represent a 4 (like veda here ) can also have the value 40 or 400 (van der Waerden 1966, p. 90). It is very remarkable, however, that after such a numerical word with the words ankair api ("in digits this is") he often writes the same number with the first nine Brahmi numbers and a small round circle for the zero (Ifrah 2000, p. 415 ). Contrary to the numerical spelling, however, he writes the digits in descending order from left to right, just as we do today. This means that our current decimal system is known to Indian scholars from 629 at the latest . Bhaskara probably did not invent it, but was the first to have no qualms about using the numerals in a scientific work in Sanskrit .

The first, however, who calculated with zero as a number and knew negative numbers, was Bhaskara's contemporary Brahmagupta .

Other work

Model of the armillary sphere described by Bhaskara .

Bhaskara wrote three astronomical papers. In 629 he commented on the verse Aryabhatiya on mathematical astronomy, precisely those 33 verses that dealt with mathematics. In it he considered indefinite first-degree equations and trigonometric formulas.

His work Mahabhaskariya is divided into eight chapters on mathematical astronomy. In Chapter 7 he gives a remarkably precise approximation formula for , namely

which he ascribes to Aryabhata . It gives a relative error of less than 1.9% (the largest deviation occurs for ). At the edge points and the approximation is exact (thus results in 0 or 1). Furthermore, relationships between sine and cosine as well as between the sine of an angle , or and the sine of an angle are listed. Parts of the Mahabhaskariya were later translated into Arabic.

Bhaskara already dealt with the statement: If a prime number, then by divisible. It was later proven for the first time by Al-Haitham, mentioned by Fibonacci , and is known today as the Wilson theorem.

Furthermore, Bhaskara formulated theorems about the solutions of what is now known as Pell's equation . So he set the problem: "Tell me, O mathematician, what is the square that multiplied by 8 and the unit results in a square?" In today's terms, the solution to Pell's equation is required. It has the simple solution , or in short , from which further solutions can be constructed, e.g. B. .


  • KS Shukla: Mahabhaskariya, Lucknow University Press 1960 (Sanskrit with commentary and English translation)
  • KS Shukla: Laghubhaskariya, Lucknow University Press 1963 (with English translation)
  • KS Shukla: Aryabhatiya of Aryabatha, with the commentary of Bhaskara I and Somesvara, Indian National Science Academy, New Delhi 1976
  • Agathe Keller: Expounding the mathematical seed. Bhaskara and the mathematical chapter of the Aryabhatiya, 2 volumes, Birkhäuser 2006 (translation of the commentary by Bhaskara I on Aryabathiya with commentary)


  • Agathe Keller: Bhaskara I., in Helaine Selin (Ed.), Encyclopedia of the history of science, technology and medicine in non western cultures, Springer 2008
  • David Pingree , article in Dictionary of Scientific Biography
  • David Pingree: Bhaskara I . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 2 : Hans Berger - Christoph Buys Ballot . Charles Scribner's Sons, New York 1970, p. 114-115 .
  • KS Shukla: Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya, Ganita, Volume 22, 1971, No. 1, pp. 115-130, No. 2, pp. 61-78, Volume 23, 1972 , No. 1, pp. 57-79, No. 2, pp. 41-50
  • B. Datta : The two Bhaskaras, Indian Historical Quarterly, Volume 6, 1930, pp. 727-736
  • B. Datta, SN Singh: History of Indian Mathematics, Lahore 1938, Asia Publishing House 1986
  • Datta, Singh: Indian Geometry, Indian Journal of the History of Science, Volume 15, 1980, pp. 21-187
  • Datta, Singh: Indian Trigonometry, Indian Journal of the History of Science, Volume 18, 1983, pp. 39-108
  • T. Hayashi, Michio Yano: A note on Bhaskara I's rational approximation to sine, Historia Scientiarum, Volume 42, 1991, pp. 45-48.
  • RC Gupta: Bhaskara I's approximation to sine, Indian J. History Sci., Volume 2, 1967, pp. 121-136.
  • RC Gupta: On derivation of Bhaskara I's formula for the sine, Ganita Bharati, Volume 8, 1986, pp. 39-41.
  • H.-W. Alten, A. Djafari Naini, M. Folkerts, H. Schlosser, K.-H. Schlote, H. Wußing: 4000 years of algebra . Springer-Verlag, Berlin Heidelberg 2003, ISBN 3-540-43554-9 , §3.2.1.
  • S. Gottwald, H.-J. Ilgauds, K.-H. Schlote (ed.): Lexicon of important mathematicians . Harri Thun publishing house, Frankfurt a. M. 1990, ISBN 3-8171-1164-9 .
  • Georges Ifrah: Universal History of Numbers , Campus Verlag, Frankfurt a. M. 1986, ISBN 3-593-34192-1 .
  • Bartel Leendert van der Waerden : Awakening Science. Egyptian, Babylonian and Greek mathematics . Birkhäuser-Verlag, Basel Stuttgart 1966.

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