Indian mathematics can be dated all the way back to 400 AD with intelligent mathematicians who have developed the way we solve math equations in geometry and number theory today. One of the geniuses who contributed greatly to not only mathematics but astronomy as well is Brahmagupta. A man way beyond his time, academically, whose claims in astronomy and mathematics is pondered upon by math historians and astronomers to this day. I will review his very vague early life, his education and work, including his somewhat controversial books, and just a couple theories he is most famous for such as the cyclic quadratic formula and development of zero.

Brahmagupta’s life begins with perplexity, there is little to no record of his early life. His father’s name was Jishnugupta but not much information on his personal life or other relatives besides an article which states, “As a young man he studied astronomy extensively. He was well-read in the five traditional siddhanthas on Indian astronomy, and also studied the work of other ancient astronomers such as Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra” (The Famous People). He was born in a village in North West Rajastan called, Bhillamala now known as Bhinmal, in 598 AD into an orthodox Shaivite Hindu family (The Famous People).

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It is presumably that his early childhood was filled with the influence from astronomy and mathematics given his future career because he spent most of his life in Bhinmal where he broadly studied astronomy and the works of famous astronomers before him. He attended the Brahmapaksha School, a major astronomy school in ancient India at the time and became an astronomer. He was the head of the Ujjain astronomical observatory, the center of mathematics where he witnessed amazing mathematicians work. (O’Connor and Robertson 1). He became a local celebrity around this time given his intellectual contributions to the newly developed studies of astronomy and mathematics and flourished as an astronomer. (The Famous People). Not a great amount is known about his personal life, even his death is a complicated subject. It is believed that his death was somewhere between 660 AD 670 AD believed 668 AD (Famous Mathematicians; Pranesachar 248; The Famous People).

In Brahmagupta’s many influential years of life, he wrote many textbooks containing theories and formulas that contribute to astronomy and mathematics (Famous Mathematicians). He wrote his first book on mathematics and astronomy in 628 called, Brāhmasphuṭasiddhānta which mentions many theories historians acknowledge but most notably the concept known as “...one of the first mathematical books to provide concrete ideas on positive numbers, negative numbers, and zero… In Brāhmasphuṭasiddhānta he developed Brahmagupta’s Formula, the formula for calculating the area of a cyclic quadrilateral given the length of the sides” (The Famous People).

Another incredible theory he proposed was the length of a year which he initially calculated to be 365 days 6 hours 5 minutes 19 seconds but later changing the value to 365 days 6 hours 12 minutes 36 seconds in his second book Khandakhadyaka, both theories come very close to our true calculations of the solar year with a value of 365 days 5 hours 48 minutes 45 seconds. (Famous Mathematicians; O’Connor and Robertson 5; The Famous People). Brahmagupta also acknowledges gravity in his first book stating, ‘Bodies fall towards the earth as it is in the nature of the earth to attract bodies, just as it is in the nature of water to flow’(Famous Mathematicians).

Although Brahmagupta mentions more mathematical theories and astronomical theories which contribute greatly to those specific subjects, it is important to list a few theories with a little more detail. “Historians believe the first ten chapters are the first version of Brahmagupta’s work and contributions to astronomy such as “…mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipse; solar eclipses; risings and settings; the moon’s crescent; the moon’s shadow; conjunctions of the planets with each other; and conjunctions of the planets with the fixed stars” (O’Connor and Robertson 1). “Through this book, he laid the foundations of the two major fields of Indian mathematics, pati-ganita (“mathematics of procedures,” or algorithms) and bija-ganita (“mathematics of seeds,” or equations) … He also introduced new methods for solving quadratic equations and gave equations to solve systems of simultaneous indeterminate equations, in addition to providing two equivalent solutions to the general quadratic equation” (The Famous People).

Brahmagupta’s infamous formula to solve for any cyclic quadrilateral should be further observed. In an online article written by a Mathematical Olympiads trainer, C. R. Pranesachar, he reviews many theories and formulas developed in Brahmagupta’s first work. One of many is his theories: “Brahmagupta gave a simple method to construct cyclic quadrilaterals with integer sides, integer diagonals and integer area. The two different (non similar) right-angled triangles with sides (a,b,c) and (x,y,z), where c and z are hypotenuses” (Pranesachar 250). Pranesachar then goes on summarizing the formula as “…Then we have a cyclic quadrilateral ABCD with the integer sides bz, cy, az, cx and integer diagonals ay+bx, ax+by and integer area 1/2(ax+by)(ay+bx))”(Pranesachar 250). Earlier it was mentioned that his works were somewhat controversial, this is because they were found to be way too similar to Heron’s Formula and Brahmagupta’s lack of continuity in his formulas (Atzema 20-22).

Another formula created by Brahmagupta has historians skeptical because of its similarity to Heron’s formula of an area of a convex cyclic quadrilateral. “Area (ABCD) = square root of (s-a)(s-b)(s-c)(s-d)…there is actually no evidence that the latter formula was ever transmitted to India during medieval times and we do not know what might have inspired Brahmagupta’s claim…To make matters even more confusing, in accordance with the earlier Indian mathematical tradition, Brahmagupta does not explicitly say that the quadrilateral has to be cyclic, although this was probably implied” (Atzema 22).

Brahmagupta was definitely a self-assured individual when it came to theorizing, he would criticize works of previous mathematicians and astronomers and in his second textbook Khaṇḍakhādyaka, he revises his previous works which clearly shows inquisitive and judgemental features. In an online article by The Famous People, it is mentioned that Brahmagupta was “bitter in criticizing the ideas advanced by rival astronomers hailing from the Jain religion” and was also one of the few contemporaries that believed the Earth was a sphere and not flat (The Famous People).

After reviewing Brahmagupta’s life and contribution to math and astronomy it is evident that without his ideas and formulas geometry and number theory would not be the way we know it today. Not only do Brahmagupta’s works astonish many people today but his knowledge in certain subjects that had not yet been discovered exemplify his intellect and genius. Not much was known about his personal life but what is known about his professional life is shown through theories and formulas and makes up for personal details missed.

## Works Cited

- Atzema, Eisso J. “From Brahmagupta to Euler: On the Formula for the Area of a Cyclic
- Quadrilateral.” BSHM Bulletin: Journal of the British Society for the History ofMathematics, vol. 30, no. 1, Jan. 2015, pp. 20–34. EBSCOhost, doi:10.1080/17498430.2014.942818.
- “Brahmagupta.” Famous Mathematicians, https://famous-mathematicians.com/brahmagupta/.
- J J O'Connor and E F Robertson, School of Mathematics and Statistics, University of St Andrews, Scotland, November 2000
- Pranesachar, C. R. “Brahmagupta, Mathematician Par Excellence.” Resonance, vol. 17, no. 3, 22 Mar. 2012, pp. 247–252., doi:10.1007/s12045-012-0023-x.
- “Who Was Brahmagupta? Everything You Need to Know.” Childhood, Life Achievements & Timeline, https://www.thefamouspeople.com/profiles/brahmagupta-6842.php.