(a) Highlight the contributions of Aryabhatta, Varahamihira and Brahmagupta in the fields of Astronomy and Mathematics. (20…

(a) Astronomy and Mathematics: Aryabhatta, Varahamihira, and Brahmagupta

Aryabhatta (Aryabhatiya, 499 CE) laid the foundations of Indian classical mathematics and astronomy. He calculated π to 3.1416, used the decimal place-value system, gave methods for square roots, series sums and the kuttaka (indeterminate equations), and composed the earliest Indian sine (jya) table at 3°45′ intervals. In astronomy he held the Earth to be spherical and explained that the apparent westward motion of the stars is due to the Earth’s own rotation — an idea not revived in Europe before Copernicus, though he himself remained geocentric — and gave correct scientific explanations of solar and lunar eclipses (the Moon’s and the Earth’s shadows) rather than mythological causes.

Varahamihira synthesized five astronomical traditions — Paulisha, Romaka, Vasishtha, Surya and Paitamaha — in his Panchasiddhantika, preserving Indian and Greco-Roman (Romaka, Paulisha) as well as Alexandrian material; his Brihat Jataka and Laghu Jataka are standard astrological works, and tradition counts him among the Navaratnas of Vikramaditya’s court. His encyclopedic work, Brihatsamhita, integrated planetary motions with meteorology, hydrology, and geography, bridging mathematical computation with natural sciences.

Brahmagupta (Brahmasphutasiddhanta, 628 CE; Khandakhadyaka, 665 CE) was the first to formalize operational rules for zero, treating negative numbers as debts and setting out algebraic operations. He gave the famous formulas for the area and diagonals of cyclic quadrilaterals, advanced the kuttaka (pulveriser) and took up the varga-prakriti problem (Nx² + 1 = y²), completed much later by Bhaskara II’s chakravala method, and established an early second-order interpolation formula in spherical astronomy.

(b) Course of Pallava-Chalukya Conflicts (6th–8th Century CE)

Driven by imperial prestige, strategic control over the fertile Krishna-Tungabhadra doab, and supremacy over the Vengi region, the Pallava-Chalukya conflict began when Chalukya Pulakeshin II invaded the Tamil country and defeated Mahendravarman I at Pullalur (c. 618–619), a campaign celebrated in Ravikirti's Aihole inscription (634); he had already seized Vengi, where his brother founded the Eastern Chalukya line. Pallava Narasimhavarman I reversed this loss in 642 CE by defeating Pulakeshin II, capturing and sacking the Chalukya capital Vatapi, and assuming the title Vatapikonda.

The struggle continued through alternating campaigns. Vikramaditya I recovered Vatapi and seized Kanchi, but Parameshvaravarman I defeated him at Peruvalanallur (c. 674). In the eighth century Vikramaditya II occupied Kanchi (his Kannada inscription is in the Kailasanatha temple) but spared its temples, and his queen Lokamahadevi built the Virupaksha temple at Pattadakal to mark the victory over Nandivarman II Pallavamalla. This protracted warfare stimulated advanced siege operations, naval logistics (demonstrated in Narasimhavarman’s Sri Lankan expeditions), and competitive architectural patronages—exemplified by the monolithic rathas of Mamallapuram and the Virupaksha temple at Pattadakal. Ultimately, continuous mutual warfare drained the material resources of both empires, enabling the Rashtrakuta Dantidurga to replace the Chalukyas (c. 753) and facilitating the eventual rise of the imperial Cholas.

(c) Role of Agraharas in Early Medieval Education

Agraharas (tax-free brahmadeya land grants endowed to Brahmins) under the Pallavas (such as the ghatikas of Kanchi), Badami Chalukyas, and Rashtrakutas functioned as prime centers of higher learning. They supported scholars who taught the Vedas, Dharmashastras, grammar, astronomy, and philosophy. These settlements maintained manuscript libraries and integrated temple-centered pedagogical spaces (mathas and salais). The Talagunda inscription records the Kadamba founder Mayurasharman going to study at the ghatika of Kanchi. Inscriptions describe the Rashtrakuta-period college at Salotgi (945 CE) with hostels for students from many regions, and the Chola-period colleges at Ennayiram (about 340 students) and Tribhuvani (about 260 students) with endowed teachers.

However, critically, Agraharas fostered institutional elitism. Education was largely confined to Brahmins and upper castes, with Sanskrit as the main medium, while crafts and practical knowledge were learnt in guilds and families. R.S. Sharma and other proponents of 'Indian feudalism' see land grants as instruments of ideological legitimation and agrarian dominance; B.D. Chattopadhyaya instead stresses their role in integrating new regions into a shared Brahmanical culture. Buddhist monasteries and Jain centres ran parallel systems of learning.