Tuesday, 22 July 2014


Brahmagupta (598-668 CE) was a prominent astronomer-mathematician from India, who hailed from the ancient town of Bhillamala, which today is the village of Bhinmal near Mount Abu close to the Rajasthan-Gujarat border. He was a contemporary of Bhāskara I.

His most important works were Brahmasphutasiddhānta and the Khaṇḍa Khādyaka, both of which deal with astronomy and mathematics. The date of composing the Brahmasphutasiddhānta was in 628 CE at the age of 30 years. Khaṇḍa Khādyaka was composed in 665 CE (when he was 67 years old). Important commentaries were written on these texts by later mathematicians such as Prithudakaswami, Lalla and Bhattotpla (see [8]). Brahmagupta made significant contributions to astronomy, algebra, geometry, numerical analysis in his works. We will discuss mainly the mathematical works since the astronomical works are too detailed and complex for a general audience.

Brahmagupta is probably most famous for his brilliant result in algebra known as Brahmagupta’s lemma (see references [7-9]). This result is a key intermediate step for getting the general solution to the equation of the sort x2-Dy2 = 1. (Quadratic algebraic equations with integral solutions). Brahmagupta solved this equation for the specific cases of D = 83 and 92. For D = 92, the solutions are x = 11512 and y = 1202. For D = 83, the solution is x = 822 and y = 92. (see reference [9], entry on Cakravala.) 
The method to solve for general D, which is known as the chakravala algorithm and which uses Brahmagupta’s lemma as a key step, was obtained later by Bhāskaracharya II and Jayadeva. This equation is today known as the Pell’s equation, even though it was solved several centuries earlier by Indian mathematicians. Other names, Indian in origin, would be more suitable which would properly reflect the origin of the problem and its original solution. Indeed, “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 ofour greatest modern analysts.’ Indian mathematical historians would like to call it the Brahmagupta–Bhāskara problem, keeping in mind that Bhāskara perfected Brahmagupta’s method of solution in the twelfth century; Bhāskara used ‘Chakra-vala,’ or a cyclic process, to improve Brahmagupta’s method by doing away with the necessity of finding a trial solution.” (see reference [8], pg. 187.)
Brahmagupta also discussed the kuttaka algorithm of Aryabhata in Brāhmasphutasiddhānta, which is used to solve linear algebraic equations with integral solutions.
Brahmagupta made significant discoveries in geometry as well. He obtained a formula for the area of a cyclic quadrilateral which appeared for the first time in the history of mathematics (encyclopedia), as well as expressions for the diagonals of a cyclic quadrilateral. In Brāhmasphutasiddhānta he gave a simple rule for forming the so-called Brahmaguptan quadrilaterals, which are cyclic quadrilaterals whose sides and diagonals are integral and whose diagonals intersect orthogonally.

The simplicity and beauty of the Indian number system is often overlooked. The ease with which it handles calculations means that Indian mathematicians were able to carry out complex calculations. Often it is not possible to calculate a function exactly. Sometimes the value of a function is given only at discrete points and its values at intermediate points is needed (interpolation). This forms the subject of numerical analysis. The need to make complex calculations and the ability to do so due to a superior number system led to forays into numerical techniques as well.  Brahmagupta went beyond linear interpolation method and obtained the equivalent of Newton-Stirling interpolation formula up to the second order. In chapter IX of Khaṇḍa Khādyaka, he introduced a new method of obtaining interpolated values of sines from a given table of sine values. Implicitly uses notions of calculus. So in fact, Brahmagupta, Bhāskara II laid the foundations of calculus, with Kerala mathematicians (mention this point in Bhāskara II and Kerala mathematicians).  This is discussed in more detail in reference [4], Vol. 1 pg. 99, as well as in reference [6]. We present the original Sanskrit verse and its English translation from the latter reference:

(Dhyāna-Graha-Upadeśa-adhyāya, 17; Khaṇḍa Khādyaka, IX, 8) whose translation is:

“Multiply half the difference of the tabular differences crossed over and to be crossed over by the residual arc and divide by 900’ (=h). By the result (so obtained) increase or decrease half the sum of the same (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.” (see reference [6].)

The actual verse is also to be found in pg. 177 of reference [5]. Reference [6] also discusses Bhaskara II’s ideas on this in more detail.

To shed more light on the above, we quote directly from [7]:

“In modern notation, the rule employed by Brahmagupta is equivalent to the formula:

up to second-order differences. So, Brahmagupta was the first mathematician in the history of world mathematics to introduce second-order difference interpolation nearly 10 centuries before the rediscovery of this formula which is presently known as the Newton-Stirling formula.” (reference [7] pg. 204). 

“In astronomy, Brahmagupta discussed the average and real motions of the planets, the problems of place-time-distance concerning the earth, sun, and planets, planetary conjunctions, and the rising and setting of celestial objects. He correctly described the phenomena of solar and lunar eclipses as being caused by the moon and earth casting shadows, on which he based his calculations. One chapter is devoted to the description and use of various astronomical instruments.” (as quoted from [8]). Also “...was not a mere theorist, he made his calculations on direct observations with the help of instruments or devices; he was in favour of making corrections on the basis of these observations. He was himself an expert observer. In his Khaṇḍa Khādyaka also he has emphasized the need of direct observation.” (as quoted from [4] Vol. I pg. 48)

 The life of Brahmagupta is very instructive in demonstrating the state of the sciences in ancient India. In stark contrast to/demolishes the modern westernized accounts of the state of science and society in ancient India. For example, in the Brāhmashuptasiddhanta, he states ([4] Vol. I pg. 48): “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.” Criticized Aryabhata’s system of astronomy....Aryabahata was the most influential. So Indian mathematics was not based on dogma. It was open to change and modification. Was a thriving and dynamic process. Also was a Vaishya...which shows that knowledge was not the exclusive preserve of the Brahmin caste, and that caste system was not birth based. He was critical of the astronomical models of earlier astronomers such as Aryabhata. It goes on to show that Indian science was not static but was open to questioning accepted views, in tune with modern scientific attitude. Thus the claim that the scientific revolution began in Europe is not entirely true. Rather, it was prevalent in India till the Islamic invasions and European colonization shattered the prosperity of the Indian society.
Bhāskara II’s Siddhantashiromani (1150) was almost entirely based on Brāhmashuptasiddhanta ([4] Vol. I pg. 48). Bhāskara II praised him as Ganakachakrachudamani, jewel among the circle of mathematicians.

[1] T. S. Bhanumurthy, A Modern Introduction to Ancient Indian Mathematics, published by New Age International, 2009.
[2] H. S. M. Coxeter and S. L. Greitzer, Geometry Revisited, publ. Mathematical Association of America, 1967.
[3] S. Goonatilake, Toward a Global Science - Mining Civilizational Knowledge, Indiana University Press, 1999.
[4] Brāhmashuptasiddhanta of Brahmagupta, in 4 volumes, Sanskrit text with translation and commentary in Hindi, and an extensive foreword in English, published by Indian Institute of Astronomical and Sanskrit Research, New Delhi, 1966. (Click here for online version of the first volume. Link opens in a new window. Links for downloading the book in various formats including pdf are on the left. The other three volumes are also available on the same site.)
[5] Khaṇḍa Khādyaka of Brahmagupta, with Sanskrit commentary of Bhattotpala and English commentary by Bina Chatterjee, 1970. (click here for pdf version online. Link opens in new window.)
[6] R. C. Gupta, Second order interpolation in indian mathematics up to the fifteenth century, Indian Journal of History of Science, 1969.
[7] T. K. Puttaswamy, Mathematical Achievements of Pre-modern Indian Mathematicians, published by Elsevier, 2012.
[8] B. S. Yaday and Man Mohan (eds.), Ancient Indian Leaps into Mathematics, Birkhäuser (Springer) 2011.

[9] Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Ed. Helaine Selin, Springer 2008.

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