18 Jul 2025
Āryabhaṭa, Bhāskarācārya, and Kaṇāda
Historically bounded contributions to astronomy, mathematics, motion, gravity, and atomism.
Āryabhaṭa and Bhāskara II were mathematical astronomers; Kaṇāda is the traditional author associated with a philosophical school. Calling all three “physicists” can be convenient in a syllabus, but their texts, aims and evidence differ from those of a modern research discipline.
Āryabhaṭa: rotation, planetary computation, and Earth
Āryabhaṭa completed the Āryabhaṭīya in 499 CE. Its mathematical astronomy explains daily stellar motion through Earth’s axial rotation and uses relative motion explicitly. It also supplies algorithms for planetary longitudes and eclipses.
His approximation to $\pi$ is encoded by
\[\pi\approx\frac{62832}{20000}=3.1416.\]The text gives Earth’s diameter as $1050$ yojanas, so this value implies
\[C\approx\pi D=3298.68\ \text{yojanas}\approx3300\ \text{yojanas}.\]The physical length assigned to a yojana is historically variable, so a uniquely precise modern kilometre conversion cannot be claimed without first fixing which unit was intended. The spherical Earth, axial rotation and numerical circumference are well grounded; full Copernican heliocentrism is not.
Bhāskarācārya: rates, falling, and planetary models
Bhāskara II (born 1114) composed the Siddhāntaśiromaṇi, whose parts treat arithmetic, algebra, planetary computation and the celestial sphere. In astronomical problems he uses tatkālika-gati, instantaneous motion, and recognizes that an instantaneous rate vanishes at an extremum. In modern notation that local idea is
\[v(t)=\lim_{\Delta t\to0}\frac{x(t+\Delta t)-x(t)}{\Delta t}, \qquad v(t_*)=0\ \text{at a smooth turning point}.\]This is an important differential-style technique, but it is not yet the general symbolic differential and integral calculus later organized around functions, derivatives, integrals and a fundamental theorem.
A passage in the astronomical work is often translated as saying that Earth has an attractive power by which objects fall toward it. It is evidence of a qualitative gravitational idea. It gives neither a universal force between all masses nor the inverse-square law $F=Gm_1m_2/r^2$, so “Bhāskara discovered Newtonian gravity” is not historically warranted.
Bhāskara refined inherited mean-motion and correction procedures for planetary positions. These were computational geocentric models, not dynamical ellipses derived from gravitation.
Kaṇāda: atomism and classification
Kaṇāda’s dates are uncertain, and the surviving Vaiśeṣika Sūtra and its doctrine have a layered history. The tradition associated with him classifies substance, quality and motion and explains perceptible matter through imperceptible paramāṇus of earth, water, fire and air.
This is an atomic model in the historical sense of a theory of indivisible material units. It is not the modern atomic model, in which nuclei and electrons are governed by quantitative electromagnetic and quantum laws. Kaṇāda’s enduring contribution is the systematic problem he posed: how can stable properties and observable change arise from enduring constituents and their relations?
Discussion