Ancient-education/Philosophers and Scholars/Brahmagupta

Brahmagupta

Brahmagupta was a great Indian mathematician and astronomer who lived in the early 7th century CE. He wrote two important books, namely, the ‘Brāhmasphuṭasiddhānta’, which explained mathematical and astronomical rules, and the ‘Khandakhadyaka’, which gave practical methods for calculations. Brahmagupta was the first Indian thinker to describe gravity as a force that pulls objects, using the Sanskrit word gurutvākarṣaṇam’. He also gave the earliest clear explanation of the quadratic formula for solving quadratic equations. His work laid strong foundations for later studies in mathematics and astronomy. ^[1]

Early life and background

Brahmagupta stated that he was born in 598 CE in Bhillamala, in the region known as Gurjaradesa, which is today Bhinmal in Rajasthan, India. At that time, the area was ruled by the Chavda dynasty under King Vyagrahamukha. He was the son of Jishnugupta and followed Hinduism, particularly the Shaivite tradition. Brahmagupta spent a large part of his life in Bhillamala, working and studying there. A later commentator, Prithudaka Svamin, referred to him as Bhillamalacharya, meaning “the teacher from Bhillamala.”.

Bhillamala was the capital of Gurjaradesa, the second-largest kingdom in Western India, covering parts of southern Rajasthan and northern Gujarat. It was also an active centre for learning, especially mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy at the time. He studied the five traditional Siddhantas and learnt from earlier scholars, such as Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin, and Vishnuchandra.

In 628 CE, at the age of 30, he wrote the Brāhmasphuṭasiddhānta, a revised and expanded version of the Brahmapaksha Siddhanta. The text has 24 chapters and 1008 verses, covering astronomy and many areas of mathematics. Later, he moved to Ujjain, where he wrote his second major work, the Khandakhadyaka, at the age of 67. Brahmagupta is believed to have died in Ujjain in 668 CE. ^[1]

The Brāhmasphuṭasiddhānta

The Brāhmasphuṭasiddhānta, written by Brahmagupta in 628 CE when he was 30 years old, is one of the most important early works in Indian mathematics and astronomy. The title means ‘The Correctly Established Doctrine of Brahma’. It is a large treatise made up of 24 chapters and 1008 verses written in the Arya metre. Although it mainly deals with astronomy, it also contains many important mathematical ideas that later influenced scholars in India and other parts of the world.

In this work, Brahmagupta explains the rules for arithmetic, algebra, geometry, and trigonometry in a clearly organised manner. His treatment of zero, positive numbers, and negative numbers is one of the most important parts of the text. He gave rules for using zero in addition, subtraction, multiplication, and division, which was a major step forward in the history of mathematics. The text also includes methods for solving quadratic equations, making it one of the earliest works to offer a clear form of the quadratic formula.

On the astronomical side, the Brāhmasphuṭasiddhānta describes planetary motions, eclipses, and methods for calculating the positions of celestial bodies. It also includes Brahmagupta’s early explanation of gravity as a force of attraction. The work became a key reference for later Indian and Islamic astronomers and mathematicians. ^[2]

The Khandakhadyaka

The Khandakhadyaka was written by Brahmagupta when he was 67 years old in 665 CE. It is another major work on Indian astronomy. Unlike his earlier text, the Brāhmasphuṭasiddhānta, which is more theoretical, the Khandakhadyaka belongs to the karana category of astronomical texts. These texts were written to help students and working astronomers perform calculations more quickly.

The Khandakhadyaka presents clear methods for calculating the positions of the Sun, Moon, and planets. It includes tables and rules that make it easier to work out planetary motions, eclipses, and timekeeping. Brahmagupta used the midnight day-count method in this book, which became very influential in later Indian astronomy. The text also explains how to find the true and mean positions of planets and use corrections to improve accuracy.

Because of its practical nature, the Khandakhadyaka became widely used for many centuries. Later scholars in India and the Islamic world studied and commented on it, and it helped shape astronomical practice across a large region. Its clear instructions, organised structure, and reliable methods make it an essential guide for students and astronomers long after Brahmagupta's time. ^[3]

Breakthroughs in Mathematics

Brahmagupta’s ideas changed the way numbers and equations were understood. His clear rules, new methods, and practical formulas laid the foundation for many areas of modern mathematics. Mentioned below are some of his major works and contributions, showing how his discoveries shaped arithmetic, algebra, geometry, and astronomical calculation.

Algebra

Brahmagupta made some of the earliest clear rules for solving algebraic equations. He provided updated methods for solving linear and quadratic equations. He explained how to isolate unknown numbers and how to use rules step by step. He also talked about solving harder equations with many unknowns and suggested the use of a method he called the “pulveriser”, which broke large problems into smaller ones. ^[3]

Arithmetic

Brahmagupta explained the basic rules of arithmetic using the Hindu–Arabic numerals. He described in detail how to add, subtract, multiply, and divide. He also explained how fractions work and how to combine them. His description of multiplication is close to today’s written method. Through his work, Indian arithmetic spread to other parts of the world, especially to the Middle East and Europe, where it became known as the ‘method of the Indians’. ^[3]

Squares and Cubes

Brahmagupta gave formulas to determine the sum of the first n squares and the sum of the first n cubes. His cube formula shows that the sum of the first n cubes equals the square of their sum. These ideas made large calculations easier, especially for astronomers and scholars who worked with long lists of numbers. His formulas remain important in number theory today. ^[3]

Zero

Brahmagupta was the first known mathematician to provide full rules for using zero as a number, not just as a symbol. He explained how zero behaves in addition, subtraction, multiplication, and division. Some rules, like dividing by zero, were not fully correct by modern standards, but his ideas were a major step forward. His work helped make zero a central part of mathematics across the world. ^[3]

Diophantine Analysis

Brahmagupta studied equations where only whole-number solutions are needed. He gave a method for creating Pythagorean triples, which are sets of numbers that form right-angled triangles. He also worked on what is now called Pell’s equation, giving a rule for finding new solutions from old ones. His method used an identity now known as “Brahmagupta’s Identity”, which allowed mathematicians to build long chains of whole-number solutions. ^[2]

Geometry

Brahmagupta made many discoveries in geometry. His most famous result is Brahmagupta’s Formula, which indicates the exact area of a cyclic quadrilateral using only the lengths of its sides. He also explained how to identify unknown sides and heights of triangles and quadrilaterals. His work included rules for diagonal lengths, triangle segments, rational triangles, and many other figures. These results were widely used in astronomy and construction. ^[2]

Brahmagupta’s Theorem

Brahmagupta’s Theorem explains a special property of cyclic quadrilaterals, where opposite angles add up to 180 degrees. He showed that when the diagonals of such a quadrilateral cross, the perpendicular from the intersection divides one diagonal into two equal parts. This idea links the shape’s geometry with its symmetry. The theorem helped later mathematicians understand the relationships inside circular figures. ^[2]

Pi

Brahmagupta used two values of pi. For everyday work, he used the simple value π = 3, which made calculations quick. For more accurate results, he used √10, which is about 3.162. This value was much closer to the real value of pi and showed his understanding of practical and exact methods. His choice of values helped astronomers work out distances and areas more easily. ^[2]

Measurements and construction

Brahmagupta explained how to make and measure many shapes. He gave rules for finding the area of triangles, quadrilaterals, and circles. He also described the volumes of solids such as pyramids, prisms, and the frustum of a pyramid. His instructions allowed builders and scholars to create shapes with given sides and angles. These methods were very useful in architecture, land measurement, and astronomy.

Trigonometry – Sine Table

In his work, Brahmagupta included a detailed sine table, which listed values needed for astronomical calculations. Instead of writing numbers directly, he used words to represent digits, as was common in Sanskrit texts. The table listed 24 sine values based on a fixed radius. These values helped astronomers track the positions of planets and stars with better accuracy for their time.

Interpolation Formula

Brahmagupta created a special formula to estimate new values of the sine function when only nearby values were known. This process, called interpolation, allowed him to fill in gaps in his sine table. His formula is an early example of second-order interpolation, similar to what is used in modern numerical maths. It made calculations smoother and more accurate for astronomers working with limited data.

The early concept of gravity

Brahmagupta described gravity as a pulling force, using the Sanskrit term gurutvākarṣaṇam to mean the attraction of objects toward the Earth. He wrote that bodies are drawn downwards and that this pull explains why things fall and why planets move in certain ways relative to the Earth. This idea was an early step towards thinking of gravity as a physical force, rather than as purely myth or fate. Though not like Newton’s later law, Brahmagupta’s description shows careful observation and a desire to explain natural motion based on a consistent idea of attraction. ^[4]

Astronomy

Brahmagupta came up with ways to figure out the positions of the Sun, Moon, and planets, giving both their "mean" and "true" longitudes. He did this by using epicycles to account for irregular motion.  He used geometry and trigonometry to come up with ways to figure out when solar and lunar eclipses would happen and when celestial bodies would rise and set.  He also addressed lunar phases and the illumination of the Moon, explaining why the visible crescent appears as it does by considering sunlight, the relative position of the Moon and Sun, and the geometry of illumination. These contributions improved Brahmagupta's astronomy, making it more accurate and useful for keeping track of time, making calendars, and predicting eclipses.^[5]

Legacy and Modern Study

Brahmagupta is remembered as one of the great early writers who brought clarity to arithmetic and algebra. His careful rules for zero and negative numbers removed much confusion and allowed later mathematicians to work more freely with signed numbers. His blend of astronomy and mathematics shows how both fields supported each other: accurate observation needs adequate calculation, and effective calculation supports clear predictions. For historians of science, his works remain a rich source of information about how people in the early medieval period studied the skies and the numbers that describe them. ^[2]

References

1. https://en.wikipedia.org/wiki/Brahmagupta
2. https://dn720706.ca.archive.org/0/items/Brahmasphutasiddhanta/Brahmasphutasiddhanta.pdf
3. https://dn790002.ca.archive.org/0/items/algebrawitharith00brahuoft/algebrawitharith00brahuoft.pdf
4. https://ia903205.us.archive.org/7/items/Brahmasphutasiddhanta_Vol_1/BSS_VOL_I_text.pdf
5. https://ia803205.us.archive.org/7/items/Brahmasphutasiddhanta_Vol_1/BSS_VOL_I_text.pdf