Classical mathematical astronomy: Varāhamihira & Brahmagupta

By the 6th century CE, the city of Ujjain (located on the prime meridian of ancient Indian geography, the madhya-rekhā) emerged as a major centre of subcontinental mathematical astronomy.¹

Thinkers such as Varāhamihira, Brahmagupta, and Bhāskara I advanced this tradition: Brahmagupta developed algebra (bījagaṇita) and defined arithmetic with negative numbers and zero, and both he and Bhāskara I worked on algorithms for indeterminate equations.

Varāhamihira and the Pañcasiddhāntikā (6th century CE)

Varāhamihira was an encyclopedist of astronomical knowledge:

• His masterwork, the Pañcasiddhāntikā (Compendium of the Five Astronomical Canons), compared five competing systems of planetary reckoning: the Sūryasiddhānta, Paulisa, Romaka, Vāsiṣṭha, and Paitāmaha.²
• He preserved invaluable historical records of cross-cultural exchanges between Indian astronomers and Greco-Roman (Yavana) mathematical tables.
• In his Bṛhat Saṃhitā, Varāhamihira compiled material on meteorology, cloud formations, water-table detection and metallurgy.

Brahmagupta and the formal arithmetic of Zero (628 CE)

Working at Bhillamāla (modern Bhinmal, Rajasthan), Brahmagupta composed the monumental Brāhmasphuṭasiddhānta (Correctly Established Doctrine of Brahma) in 628 CE at age 30:

1. The Laws of Zero and Negative Numbers: Brahmagupta gave one of the earliest known systematic sets of rules for positive numbers (dhana / fortune), negative numbers (ṛṇa / debt), and zero (śūnya): In summary, his rules state that the sum of two positives is positive and of two negatives negative; the sum of a positive and a negative is their difference; the product of two negatives is positive, of a positive and a negative negative, and of zero and any number zero (BSS XVIII.30–35).³

2. Indeterminate Equations of the Second Degree (Vargaprakṛti): Brahmagupta formulated an ingenious method to find integer solutions for equations of the form:

   Nx² + 1 = y²

   (Later mistakenly known in Europe as “Pell’s equation”). He invented the Bhāvanā composition lemma, enabling one solution to generate infinite subsequent integer solutions — a milestone in algebraic number theory.⁴

3. Brahmagupta’s Formula for Cyclic Quadrilaterals: He gave a rule that for a quadrilateral with side lengths a, b, c, d inscribed in a circle, the area A is given by:

   A = √[(s − a)(s − b)(s − c)(s − d)]

   Here, s = (a + b + c + d) / 2 is the semi-perimeter, or half the total length of the sides.⁵

Bhāskara I and rational trigonometric approximations (c. 600 CE)

Bhāskara I, a follower of Āryabhaṭa’s school, developed an accurate algebraic rational formula for computing the sine of an angle without tables:⁶

sin(x) ≈ 16x(π − x) / [5π² − 4x(π − x)], for 0 ≤ x ≤ π.

This rational formula exhibits a maximum relative error of less than 1.9% across the entire range from 0° to 180°.

Four terms to carry forward

What this chapter does not claim

1. That Brahmagupta solved division by zero completely; he posited 0/0 = 0, which modern calculus defines as undefined / indeterminate.
2. That Indian astronomers worked in complete isolation from the outside world; Varāhamihira openly acknowledged studying Greco-Roman tables alongside indigenous models.
3. That mathematical astronomy was purely secular; it was closely integrated with calendarial astrology (Jyotiṣa), though mathematical astronomy (Siddhānta) was treated as a distinct rigorous science.
4. That ancient algebraic methods were written in modern symbolic notation; Indian algebra was rhetorical and syncopated, composed in metrical Sanskrit verse.

Sources and further reading

Version 0.4 · September 2026 · Editorial orientation · Reading list provided · Not expert-reviewed. Corrected after an editorial audit in September 2026.