Jyotihshastra: Astral and Mathematical Literature c5.s1.9

c5.s1.9

Bhāskara’s contemporary, Brahmagupta, devotes two *adhyāyas* of his BSS to mathematics. Chapter 12⁷ is the earliest extant formal treatise on arithmetic in Sanskrit. In it he gives succinct rules for twenty arithmetical operations (*parikarma*) and eight determinations (*vyavahāra*). These are listed by Pṛthūdakasvāmin; the *parikarmāṇi* are: addition (*saṅkalita*), subtraction (*vyavakalita*), multiplication (*pratyutpanna*), division (*bhāgahāra*), square (*varga*), square-root (*vargamūla*), cube (*ghana*), cube-root (*ghanamūla*), five⁸ rules for operating with fractions (*pañcajātayaḥ*), the rule of three (*trairāśika*), the inverse rule of three (*vyastatrairāśika*), the rules of five, seven, nine, and eleven terms, and barter (*bhāṇḍapratibhāṇḍa*), while the *vyavahārāḥ* are: mixtures (*miśraka*), series (*śreḍhī*), plane geometry (*kṣetra*), (solid geometry, subdivided into) excavations (*khāta*), stacks (*citi*), sawn lumber (*krākacika*), and mounds of grain (*rāśi*), and shadow-problems (*chāyā*). Of special interest is his exhaustive investigation of triangles and quadrilaterals—mostly right triangles and cyclic quadrilaterals (BSS 12, 21—39). Chapter 18 is devoted to algebra, here denominated kuṭṭaka after the pulverizer;⁹ it is illustrated by exemplary problems (udāharaṇa). Among the topics discussed, in addition to the pulverizer itself, are the mathematics of zero and of surds, quadratic equations, equations with several unknowns, and indeterminate equations of the second degree (vargaprakṛti). Brahmagupta’s method for solving vargaprakṛti equations is not universally applicable. There is no convincing evidence to support the theory that Brahmagupta had a Greek source.¹⁰