Brahmagupta

Brahmagupta (598 – 668 CE) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the Khandakhadyaka ("edible bite", dated 665), a more practical text.

Quotes

• The sum of two positive quantities is positive; of two negative is negative; of a positive and a negative is their difference; or, if they are equal, zero. The sum of zero and negative is negative; of positive and zero is positive; of two zeros is zero (31).
• In subtraction, the less is to be taken from the greater, positive from positive; negative from negative. When the greater, however, is subtracted from the less, the difference is reversed. Negative taken from zero becomes positive; and positive [taken from zero] becomes negative. Zero subtracted from negative is negative; from positive, is positive; from zero, is zero. When positive is to be subtracted from negative, and negative from positive, they must be thrown together (32-33).
• The product of a negative quantity and a positive is negative; of two negatives, is positive; of two positives, is positive. The product of zero and negative, or of zero and positive, is zero; [the product] of two zeros, is zero. (34).
• Positive, divided by positive, or negative by negative, is positive. Zero, divided by zero, is zero. Positive, divided by negative, is negative. Negative, divided by positive, is negative. Positive, or negative, divided by zero, is a fraction with that for denominator: or zero divided by negative or positive. (35-36).
• The square of negative or positive is positive; of zero, is zero. The root of a square is such as was that from which it was raised [i.e. either positive or negative]. (37).
  • Brahmagupta, Brahmasphutasiddhanta quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• In a linear equation in one unknown, the difference of the two known terms taken in the reverse order, divided by the difference of the coefficients of the unknown (is the value of the unknown).
  • Brahmagupta in his Brahmasphutasiddhanta
  • Datta and Singh, History of Hindu Mathematics
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• Of the unknowns, their squares, cubes, fourth powers, fifth powers, sixth powers, etc., addition and subtraction are performed of the like; of the unlike they mean simply their statement severally.
  • Datta and Singh, History of Hindu Mathematics
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• Multiply half the difference of the tabular differences crossed over and to be crossed over by the residual arc and divide by 900’. By the result so obtained increase or decrease half the sum of the (two) differences, according as this (semi-sum) is less or greater than the difference to be crossed over. We get the true functional differences to be crossed over.
  • The above verse gives the formula for second order interpolation and is to be found in the Khandakhadyaka. This was the first time in the history of mathematics that a second-order formula for interpolating values of a function was discovered. The formula discovered by Brahmagupta in the above verse is today known as the Newton-Stirling formula and was discovered in Europe nearly ten centuries later
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• The old calculations dealing with planets based on the system of Brahma have become erroneous in course of past ages and therefore I, the son of Jishnugupta would like to clarify them.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

About

• Florian Cajori, the noted historian, summed up the matter in an extraordinarily suggestive manner: The perversity of fate has willed it that the equation y2 = nx2 + 1 should now be called Pell’s Problem, while in recognition of Brahmin scholarship it ought to be called the “Hindu Problem.” It is a problem that has exercised the highest faculties of some of our greatest modern analysts. Indian mathematical historians would like to call it the Brahmagupta–Bhaskara problem, keeping in mind that Bhaskar perfected Brahmagupta’s method of solution in the twelfth century; Bhaskara used “Chakravala”, or a cyclic process, to improve Brahmagupta’s method by doing away with the necessity of finding a trial solution.
  • (Bhattacharyya 2011: 187). Bhattacharyya, R. K. ‘Brahmagupta: The Ancient Indian Mathematician’. In Ancient Indian Leaps into Mathematics, edited by B. S. Yadav and Man Mohan, pp. 185-192. Birkhäuser, 2011.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189