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Sign in. Verified email at sarma. Articles Cited by. From China to Paris: years transmission of mathematical ideas , , Proceedings of the Indian History Congress 55, , Indian journal of history of science 29 4 , , Indian Journal of History of Science 34 2 , , Ancient Indian leaps into mathematics, , Indian journal of History of Science 34 4 , , Supplement to Studies in History of Medicine and Science 10, , Indian Journal of History of Science 35 2 , , Studies in the History of Medicine and Science 13 1 , , Studies on Indian culture, science and literature: Being Prof.
Sri Venkateswara University Oriental Journal 41 1—2 , , Articles 1—20 Show more. Help Privacy Terms. Rule of three and its variations in India SR Sarma From China to Paris: years transmission of mathematical ideas , , A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself. Later, Brahmagupta moved to Ujjain , which was also a major centre for astronomy.
Brahmagupta lived beyond CE. He is believed to have died in Ujjain. Brahmagupta directed a great deal of criticism towards the work of rival astronomers, and his Brahmasphutasiddhanta displays one of the earliest schisms among Indian mathematicians. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself.
Ancient Indian leaps into mathematics / B.S. Yadav, Man Mohan, editors - Details - Trove
In Brahmagupta's case, the disagreements stemmed largely from the choice of astronomical parameters and theories. The historian of science George Sarton called him "one of the greatest scientists of his race and the greatest of his time. Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations. Lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka. Expeditions were sent into Gurjaradesa. The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed the attacks.
The court of Caliph Al-Mansur — received an embassy from Sindh, including an astrologer called Kanaka, who brought possibly memorised astronomical texts, including those of Brahmagupta. Brahmagupta's texts were translated into Arabic by Muhammad al-Fazari , an astronomer in Al-Mansur's court under the names Sindhind and Arakhand. An immediate outcome was the spread of the decimal number system used in the texts.
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The mathematician Al-Khwarizmi — CE wrote a text called al-Jam wal-tafriq bi hisal-al-Hind Addition and Subtraction in Indian Arithmetic , which was translated into Latin in the 13th century as Algorithmi de numero indorum. Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout the world. Al-Khwarizmi also wrote his own version of Sindhind , drawing on Al-Fazari's version and incorporating Ptolemaic elements.
Indian astronomic material circulated widely for centuries, even passing into medieval Latin texts. Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta ,. The difference between rupas , when inverted and divided by the difference of the [coeffecients] of the unknowns, is the unknown in the equation.
The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted. He further gave two equivalent solutions to the general quadratic equation. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square.
He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient. In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns. Subtract the colors different from the first color. If there are many [colors], the pulverizer [is to be used].
Like the algebra of Diophantus , the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms. The four fundamental operations addition, subtraction, multiplication, and division were known to many cultures before Brahmagupta.
This current system is based on the Hindu Arabic number system and first appeared in Brahmasphutasiddhanta. Brahmagupta describes the multiplication as thus "The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier". In Brahmasphutasiddhanta, multiplication was named Gomutrika. In the beginning of chapter twelve of his Brahmasphutasiddhanta , entitled Calculation , Brahmagupta details operations on fractions.
The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.
The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]. Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice. In chapter eighteen of his Brahmasphutasiddhanta , Brahmagupta describes operations on negative numbers.
He first describes addition and subtraction,.
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The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero. But his description of division by zero differs from our modern understanding:.
A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero.
Ancient Indian Leaps into Mathematics
That of which [the square] is the square is [its] square-root. In chapter twelve of his Brahmasphutasiddhanta , Brahmagupta provides a formula useful for generating Pythagorean triples :. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.
Also, if m and x are rational, so are d , a , b and c. A Pythagorean triple can therefore be obtained from a , b and c by multiplying each of them by the least common multiple of their denominators. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.
The nature of squares: The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas. The key to his solution was the identity, .
The solution of the general Pell's equation would have to wait for Bhaskara II in c. Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral.
The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral. Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case. Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle's base is divided into by its altitude:.
The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment. He further gives a theorem on rational triangles.
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