Brahmagupta

Brahmagupta was a significant Indian mathematician and astronomer who lived during the medieval era and made several indispensable contributions to various fields of mathematics and astronomy throughout his lifetime. Although many of the specific details of Brahmagupta’s birthplace are unknown, most scholars agree that he was born in 598 CE somewhere in northern India (Joseph 41-42; Waghmare et al. 1). One hypothesis is that he was born in Bhinmal (a city in the Rajasthan Sate of Northern India which was quite powerful during that time period) but no one knows for sure. One thing is certain however; the time Brahmagupta was born would play a larger role in defining his later works than the place he was born. [The 6^th century BCE was characterised by a rise in philosophical movements that challenged Hindu Orthodoxy. These groups, which were labeled heterodox by orthodox Hindus, generally challenged the Vedas and the Varna (class) system. As time progressed the number of heterodox philosophies increased and by the 6^th century CE they had 1200 years to spread and flourish throughout India. ] As an orthodox Hindu, Brahmagupta was influenced heavily by his religious beliefs and was opposed to those held by the various heterodox darsanas (viewpoints). In particular, he was intrigued by the Hindus’ Yuga system (which measures the ages of humanity) and opposed to the Jains’ cosmological views, allowing the former to greatly influence his own ideas and harshly condemning the latter (Waghmare et al. 1). The influence of orthodox Hinduism on his work did not end here.

Brahmagupta went even further in his critique of heterodox ideas when he attacked Aryabhata. Brahmagupta refuted Aryabhata’s heterodox idea that the earth is a spinning sphere (Waghmare et al. 1). The influence of religion on Brahmagupta and his works went even farther than this however. Brahmagupta’s main work Brahmasphutasiddhanta (The Correctly Established Doctrine of Brahma) which is a mathematical treatise of invaluable quality is a paradigmatic example of the extent of which religious views influenced Brahmagupta [This demonstrates Brahmagupta’s religious affiliations with Hindu orthodoxy because Brahma is believed to be the creator deity in the Hindu tradition] (Waghmare et al. 1-2). Although religious beliefs played a profound role in influencing Brahmagupta, they were by no means the only stimulus instigating his mathematical and astronomical works. As a young man, Brahmagupta was a disciple of Varahmihir, a great astronomer of the time, who had written extensively. It is said that Brahmagupta read all Varahmihir’s works, made commentaries on them, and later proved many unproved results (Waghmare et al. 1). This launched Brahmagupta’s career in mathematics and astronomy.

As mentioned earlier, Brahmagupta’s main work Brahmasphutasiddhanta was a very influential mathematical treatise influenced by orthodox Hinduism. Interestingly, this biased approach did not compromise the quality of the work entirely. In fact, R.V. Waghmare et al. describes his work as possessing mathematical ideas of “exceptional quality” and claims that it should be considered one of the greatest works of the early period “not only of India, but also of the World” (Waghmare et al. 2). The text’s incredible breadth and depth has made invaluable contributions to geometry, arithmetic, algebra, number theory, as well as astronomy. Since the text was later translated into Arabic around 771 CE it also played a profound role in the scientific awaking of the Arab Empire and had a considerable influence on Islamic mathematics and astronomy (Waghmare et al. 2). This work also had a profound impact within India. In chapters twelve and eighteen, Brahmagupta established two major fields of Indian mathematics: “mathematics of procedures” (algorithms) and “mathematics of seeds” (equations/algebra), which are still studied to this day (Waghmare et al. 2-3).

Interestingly, this is not the only text that Brahmagupta wrote. In fact, the Brahmasphutasiddhanta published in 628 CE was his second, albeit most important, work. His first work Cademekela was written in 624 CE. His third and fourth books Khandakhadyaka and Durkeamynarda were published in 665 CE and 672 CE respectively. Collectively, these texts are all extremely influential in many fields of mathematics. For instance, Brahmagupta’s work on arithmetic revolutionized the field. In fact, Brahmagupta is described as having a better understanding of number systems and place value than any of his contemporaries. In particular, Brahmagupta had a profound understanding of the number zero. While the number had been used to distinguish between numbers since ancient times (i.e. people used it to distinguish between numbers like 1, 10, and 100) it had never been considered an arithmetic entity in its own right. In other words, no one ever tried to do addition, multiplication, subtraction, or division with zero prior to Brahmagupta (Waghmare et al. 3-4). For this reason, Brahmagupta is credited with the discovery of the number zero (see Boyer 241-245). He did not stop here however. In fact, he went even further and extended arithmetic to the negative numbers and ended up formulating many of the rules that mathematicians still hold to be true today, with the exception that he allowed division by zero. Although phrased quite differently, Brahmagupta established these familiar rules of arithmetic: the product/quotient of similar signs is positive while the product/quotient of different signs is negative. He said that zero times anything is zero and that a number divided by zero is that number over zero, with the exception that zero divided by zero is zero (Waghmare et al. 3).

Next, the Brahmasphutasiddhanta moved onto algebra. Many algebraists believe that Brahmagupta’s most important contribution to the fields of algebra and number theory is his work done on Pell’s Equation (Waghmare et al. 6). Pell’s equation is the relation Nx² – 1 = y² where N is a constant and solutions take the form (x, y). Using what is today referred to as the Euclidean algorithm but known to contemporaries as the “pulveriser,” Brahmagupta broke Pell’s equation into several smaller equations (Waghmare et al. 6). His solution of the equation hinged on a generalization of the work of Diophantus, which is a long and complicated formula that is very important in the study number theory [Diophantine equations is a branch of number theory that concerns equations that only accept integer solutions] (Waghmare et al. 6-7). Unfortunately, this was not sufficient. With all the effort Brahmagupta put into studying Pell’s equation he could not generalize his results to an arbitrary constant N. Rather, he only proved a few specific cases and the general solution would not come until much later when Bhaskarall would prove it in 1150 CE. (Waghmare et al. 6-8).

In addition to these contributions, Brahmagupta also made contributions to the study of linear and quadratic equations. Giving an algorithm for what is equivalent to the quadratic formula which is used to solve equations of the form ax² + bx + c = 0 and it is believed that Brahmagupta may have been the first to realize the quadratic has two solutions. However, he went much farther than this. He also gave solutions to multiple variable quadratics of the form ax² + c = y² (Waghmare et al. 7). Another interesting result is known as the Brahmagupta-Fibonacci Identity. This identity basically asserts that sum of two squares is closed under multiplication, that is when you multiply a sum of two squares with another sum of two squares you will always get a sum of two squares. This is an incredibly powerful result that has had a profound impact on number theory especially when coupled with other results (Boyer 241-243; Waghmare et al. 9).

Despite all Brahmagupta’s magnificent achievements in these areas of mathematics, they seem almost insignificant when compared to his work in geometry. Unfortunately, many of his achievements in this field are ignored as credit was often given to Europeans due to the dominant Eurocentric attitude of the time (Waghmare et al. 8-9). One example of this is what is widely known as Ptolemy’s Theorem. This theorem can be used to find the diagonals of cyclic quadrilaterals (four sided figures whose vertices lie on a circle). Interestingly, Brahmagupta discovered and proved this theorem independently unaware of Ptolemy’s work (Waghmare et al. 8). Another example is Brahmagupta’s work on right angle triangles. Many of the results he proved were later credited to the European mathematicians Fibonacci in the 13^th century BC and Vieta in the 16^th century BC (Waghmare et al. 8-9). This does not mean that he is completely unrecognized though. In fact, “Brahmagupta’s Formula” is the name given to the formula used in Euclidean geometry to find the area of any quadrilateral when the side lengths are given and some of the interior angles. There is also a major theorem which bears Brahmagupta’s name. Brahmagupta’s Theorem states that if a cyclic quadrilateral is also orthodiagonal (has perpendicular diagonals) then if a line is drawn perpendicular to point of intersection of the diagonals it will bisect the opposite side (Waghmare et al. 9-10). Finally, Brahmagupta’s contributions in geometry include a study of triangles. His work dealt primarily with the relationships between the base of a triangle, the triangle’s altitude, and the side lengths of the triangle. In this study he also estimated the value of pi to be approximately three. Even though his estimation was incorrect he was close (Waghmare et al. 9-10). His final work with triangles concerned Pythagorean triples. These are sets of three numbers that satisfy the Pythagorean Theorem.

While Brahmagupta is also known for being an astronomer, he did not write as extensively on astronomy as he did on mathematics. Whatever he discovered in astronomy was often a consequence of his mathematics (Boyer 243-245; Waghmare et al. 11-12). In other words, he used logical mathematical reasoning to prove astronomical ideas. For instance, Brahmagupta reasoned that the sun was farther away from earth than the moon. Scriptural teachings supported the idea that the sun was closer to the earth than the moon was so this was revolutionary. He reasoned, however, that the moon is closer because of the way the sun illuminates it in cycles of waning and waxing (Boyer 221-223; Joseph 24-27). Although it may seem minor, Brahmagupta’s work in astronomy played a major role in the scientific awakening of Baghdad and the Arabic empire. When Brahmagupta’s Brahmasphutasiddhanta was translated into Arabic it forever changed the empire and gifted them with wonderful new mathematical and astronomical ideas that led to a full scale scientific revolution (see Joseph 22-27; Boyer 221-223, 241-245).

Reference and Further Recommended Reading

Boyer, Carl B (1968) A History of Mathematics. New York: John Wiley & Sons, Inc.

Joseph, George Gheverghese (2009) A Passage to Infinity: Medieval Indian Mathematics from Kerala and its Impact. New Delhi, India: SAGE Publications India Pvt Ltd.

Waghmare, R.V., Avhale P.S., and Kolhe S.B. (2012) “The Great Mathematician Brahmagupta” Golden Research Thoughts. Volume 2, Issue 1. (July 2012)