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Brahmagupta

Brahmagupta (c. 598 – c. 668 CE)

Brahmagupta, born in 598 CE at Bhillamāla (modern Bhinmal in Rajasthan, India), was an eminent Indian mathematician and astronomer. He lived at a time when Gurjaradesa (modern Rajasthan–Gujarat) was a prominent scholarly region. He later relocated to Ujjain, a major center of astronomical learning, where he carried out much of his work until his presumed death around 668 CE in Ujjain, Chalukya Empire.

He authored two major treatises:

Brāhmasphuṭasiddhānta (BSS) (628 CE) — a comprehensive theoretical astronomy and mathematical work comprising 24 chapters in 1008 verses.
Khandakhadyaka (665 CE) — a more practical manual for astronomical calculations.

Early Concept of Gravity

In 628 CE, Brahmagupta described a force akin to gravity, using the Sanskrit term gurutvākarṣaṇam, stating:

“All heavy things fall down to the earth by a law of nature … the nature of the earth is to attract and to keep things … and never rise upwards from the earth.”

Brahmagupta

This early qualitative understanding of Earth's attraction was a remarkable anticipation of gravitational theory, though not in Newtonian form.

Mathematical Contributions

Algebra & Quadratic Formula

Brahmagupta gave the first clear description of the quadratic formula, written in verse. In modern notation:
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He also solved linear equations and Pell-type equations using algorithms similar to the Euclidean method. His algebra was syncopated, like Diophantus.

Zero and Negative Numbers

In Chapter 18 of BSS, Brahmagupta laid down rules for zero and negative numbers:

Debt minus zero is debt; fortune minus zero is fortune.
a/0 is undefined, but he stated 0/0=00/0 = 0, which is incorrect by modern standards.

Series & Identities

He gave formulas for the sum of squares and sum of cubes, and his identity for integer solutions of Pell's equation is a cornerstone of number theory.

Geometry

He derived the area of a cyclic quadrilateral:
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He used π ≈ √10 (~3.1622), and approximated it pragmatically as 3.

Trigonometry

He created a table of sines with radius 3270 and developed a second-order interpolation formula.

Astronomy

Brāhmasphuṭasiddhānta (628 CE)

A 24-chapter theoretical treatise detailing:

Planetary longitudes, diurnal motion, eclipses
Lunar phases and planetary conjunctions
Rejected the incorrect claim that the Moon is farther than the Sun

Khandakhadyaka (665 CE)

A practical manual for astronomical computation, used by students and practitioners.

Legacy and Influence

Translated into Arabic as Sindhind under the Abbasid Caliphate
Influenced al-Khwarizmi, al-Fazari, and later European algebra
Revered by Bhāskara II as "gem of the circle of astronomers"

George Sarton called him “one of the greatest scientists of his race and the greatest of his time.”

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Conclusion

Brahmagupta revolutionized mathematics and astronomy:

Defined and used zero as a number
Developed a quadratic formula, rules for negatives
Formulated the cyclic quadrilateral area
Proposed a gravity-like force