The classical era was one of thriving innovation, and saw an explosion of manuja grantham : siddhantas, bhaashyaas, vartikaas (explanations of commentaries), karanas, tantraa, and novelties like vaakya-panchaangas, a surprising number of which have been preserved, edited, published and some even translated into English in the last few centuries. Criticism, correction, observation, refinement, innovation marked this period of several centuries and across various geographies.

An interesting aside for an economist, is the variety of currencies and coins (dinara, paNaa, kaarshapaaNa, puraaNa, svarNaa) and weights (pala, krosha) and measures (angula, hasta) discussed in the various books.

The primary focus is on astronomy, but every siddhanta discusses principal, interest, compounding, rate of growth, and such monetary calculations also.

Mahavira

Mahavira, the Jain mathematician who composed Ganita Saara Sangraha wrote the first mathematics book, shorn of astronomy.

The structure of his book is that first two or three stanzas in each chapter explain an algorithm or formula, and the rest of the stanzas are problems of that type to be solved by the reader.

His use of Jaina symbols, temples, methods of worship, calculations etc are singular hallmarks of the book.

Mahavira revels in several types of fractions: bhaaga (simple fraction), prabhaaga( fractions of fractions), bhaagaabhaaga (complex fractions), and so on. For example, one problem posed is below:

दिवसैसत्रिभिस्सपादैरयोजनषट्कं चतुर्थभागोनम्

गच्छति यः पुरुशोसौ दिनयुतवर्षेण किं कथय ॥ ३ ॥

divasais-tribhis-sa-paadair-yojana-shaTkam caturta-bhaaga-unam

Gacchati yaH purusho-asau dina-yuta-varsheNa kim kathaya

Translation: The man (purusha) who (asau) walks (gacchati) quarter (caturtha-bhaaga) less (unam) than six (shaTka) yojanaas in three (tribhi) and quarter (paadai) days (divasau), tell (kataya) how much (kim) he walks in a day (dina) and (yuta) a year (varsha).

Bhaskaracharya

The Lilavati of Bhaskara, author also of Siddhanta Siromani, is famous even to those unfamiliar with mathematics, as an example of beautiful poetry, and has a popular legend around it.

Like Mahavira, Bhaskara tossed in several examples from daily life to pose mathematics problems, and like Varahamihira, he reveled in his poetic talents.

Lilavati is the usually only mathematics book that Sanskrit dictionaries quote. It inspired innumerable commentaries, over centuries, translation into multiple languages and became the standard textbook of Indian mathematics.

Bhaskara corrected Aryabhata’s wrong formula for the volume of a sphere, which escaped even Brahmagupta (who corrected Aryabhata’s wrong formula for volume of a tetrahedron).

He also gave correct volumes for surface area of a sphere. His metaphor of a net covering a ball (kandukasya jaalam), for sphere volume hints that he had stumbled upon the germ of the idea of infinetismals and calculus. But these fields would only develop in later centuries, in Kerala.

Bhaskara also introduced the concept of kha-hara (a number divided by zero) for infinity (not just the philosophical ananta (endless).

Bhaskara was also the among the earliest to provide proofs of some of his derivations, and not leave it to commentators, or only teach students. After brief explorations by Pingala and Varahamihira, Bhaskara also explored permutations and combinations.

By Bhaskara’s time, algebra had developed into an advanced state. He acknowledges that he built on the works of his predecessors Sridhara and Padmanabha.

Historical perspective

Indian mathematicians were using irrational square roots for a thousand years and sines and cosines for several centuries before discovering negative numbers. The inspiration for negative numbers comes from commerce and the notion of debt, not any religious philosophy.

It needed six centuries and a Bhaskara to correct Aryabhata’s sphere volume mistake. Bhaskara realised division by zero yields infinity, but didn’t fully grasp its consequences.

From the finite series of Aryabhata to the infinite series of Virasena took only two centuries. They discovered infinite series summed up to a finite number for six centuries before questioning it.

Just as the steam engine was invented a century before the much simpler bicycle, the history of mathematics is replete with examples of complex concepts being discovered before much simpler concepts.

Astronomy inspired extraordinary mathematics, but also frequently fooled and misled the greatest of mathematicians.