The Role of Commentaries in Preserving and Advancing Indian Astronomy and Mathematics

Foundational works on Indian astronomy and mathematics, such as the Aryabhatiya and Brahmasphutasiddhanta, are composed in a concise, sutra-like form, necessitating commentaries for elaboration. These commentaries play a pivotal role in interpreting, preserving, and transmitting knowledge across generations, ensuring the continued relevance of these texts. By introducing new observations and methods, commentators foster the evolution of the field. Additionally, they simplify complex ideas, making them accessible and effective teaching aids. A notable example is Nilakantha Somayaji’s commentary on the Aryabhatiya, which provides elegant proofs, detailed explanations, and geometrical descriptions, serving as an exemplary resource for teaching.

Adding an intercalary month in a leap-year to synchronize solar and lunar years has been a matter of discussion in many cultures in different contexts. The cultures that have been influenced by ancient Indian astronomical traditions are not exceptions and somewhat similar discussions are found in the Buddhist literature in Pali language as well. Lunar month has a major role to play in the Buddhist calendric system, especially in the context of ecclesiastical performances and observing rainy retreat. The rainy retreat is a mandatory precept for high-ordained monks and nuns. Therefore, it is recommended for monks and nuns to have some knowledge of astronomy in order to practice these customs of religious importance and this has been the tradition for many centuries in Sri Lanka. In the 19th cent. CE 12 monks in the Koṭṭe-Śrī-Kalyāṇi-Dharma-Mahā-Saṃghasabhā, one of the fraternities of Buddhist monk community, were of the view that the intercalary month, when it is to be added, should be added before the regular Āsāḷha (Skt. Āṣāḍha = July-August) month and that it should be counted for the Vassāna (Skt. Varṣā = rainy) season which contradicts the custom that had been in practice before. This eventually led to an intellectual debate between the proponents and opponents of this claim and resulted in a few literary works. We will examine this main phenomenon of the debate beginning from the base sources of the Pali cannon, commentaries, sub-commentaries, and the literature pertaining to this particular debate.

The Saṅgīta-ratnākara (Ocean of Music) of Śārṅgadeva (c. 1225 CE) is a classical 13th century text on Indian musicology in Sanskrit. The text not only deals with various aspects of music, but also discusses several combinatorial techniques for generating and tabulating musical patterns and rhythms, some of which are yet to be properly studied by modern scholars. The fifth chapter of the Saṅgīta-ratnākara, known as the Tālādhyāya, deals with (among other things) the generation of rhythms (tālas) by making use of basic rhythmic components known as tālāṅgas. In this context, Śārṅgadeva describes the construction of the saṃyoga-meru (Hill of Combinations), which is a tool that tabulates the total number of possible tālas of a given length through the combination of one or more tālāṅgas. In this paper, we will discuss the underlying mathematical rationale employed by Śārṅgadeva in the construction of the saṃyoga-meru, drawing from the insightful Sanskrit commentaries of Siṃhabhūpāla (c. 1350 CE) and Kallinātha (c. 1420 CE).

In India, magic squares have been known for more than two millennia and have been thoroughly studied and well presented in the last millenia alone based on the extant texts. Nārāyaṇa Paṇḍita has dedicated the entire fourteenth chapter of his Gaṇitakaumudi (c. 1356 CE) to Bhadragaṇita where he systematically introduces the principles governing the construction of magic squares and also details general methods for the construction of different forms of magic squares. The focus of this paper is to present a thorough study of the various algorithms for constructing normal and pandiagonal magic squares of even and odd order using techniques propounded by Nārāyaṇa Paṇḍita in Gaṇitakaumudi. Earlier studies by a few scholars starting with Dutta and Singh have presented select algorithms by showing how consecutive pairs get placed in horse-moves. Whereas, in our presentation, we shall demonstrate that the construction of the entire square can be made by employing only horse-moves. Besides presenting this algorithm in modern parlance, in this talk we shall also discuss various properties exhibited by these squares along with their proofs and further analysis.