History of Hindu Mathematics: A Source Book v2.c17.s8.80

v2.c17.s8.80

**Elimination of the Middle Term.** The method of solving the quadratic was known amongst the Hindu algebraists by the technical designation *madhyamâharaṇa* or "The Elimination of the Middle" (from *madhyama* = middle and *âharaṇa* = removal, or destroying, that is, elimination). The origin of the name will be easily found in the principle underlying the method. By it a quadratic equation which, in its general form, contains three terms and so has a middle term, is reduced to a pure quadratic equation or a simple equation involving only two terms and so having no middle term. Thus the middle term of the original quadratic is eliminated by the method generally adopted for its solution. And hence the name. Bhâskara II has observed, "It is also specially designated by the learned teachers as the *madhyamâharaṇa*. For by it, the removal of one of the two² terms of the quadratic, the middle one, (takes place)."³ The name is, however, employed also in an extended sense so as to embrace the methods for solving the cubic and the biquadratic, where also certain terms are eliminated. It occurs as early as the works of Brahmagupta (628).¹