Indian Mathematics - The Written Tradition: Prose Commentary

The Written Tradition: Prose Commentary

With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation.

"India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least the fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally."

The earliest mathematical prose commentary was that on the work, Āryabhaṭīya (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the Āryabhaṭīya was composed of 33 sūtras (in verse form) consisting of mathematical statements or rules, but without any proofs. However, according to (Hayashi 2003, p. 123), "this does not necessarily mean that their authors did not prove them. It was probably a matter of style of exposition." From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations (upapatti). Bhaskara I's commentary on the Āryabhaṭīya, had the following structure:

• Rule ('sūtra') in verse by Āryabhaṭa
• Commentary by Bhāskara I, consisting of:
  • Elucidation of rule (derivations were still rare then, but became more common later)
  • Example (uddeśaka) usually in verse.
  • Setting (nyāsa/sthāpanā) of the numerical data.
  • Working (karana) of the solution.
  • Verification (pratyayakaraṇa, literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favored by then.

Typically, for any mathematical topic, students in ancient India first memorized the sūtras, which, as explained earlier, were "deliberately inadequate" in explanatory details (in order to pithily convey the bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (i.e. boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta (fl. 7th century CE), to characterize astronomical computations as "dust work" (Sanskrit: dhulikarman).