History of Hindu Mathematics: A Source Book v2.c17.s15.1

v2.c17.s15.1

**Cyclic Method.** It has been just shown that the most fundamental step in Brahmagupta's method for the general solution in positive integers of the equation $Nx² + 1 = y²,$ where N is a non-square integer, is to form an auxiliary equation of the kind $Na² + k = b²,$ where a, b are positive integers and k = ± 1, ± 2 or ± 4. For, from that auxiliary equation, by the Principle of Composition, applied repeatedly whenever necessary, one can derive, as shown above, one positive integral solution of the original Square-nature. And thence, again by means of the same principle, an infinite number of other solutions in integers can be obtained. How to form an auxiliary equation of this type was a problem which could not be solved completely and satisfactorily by Brahmagupta. In fact, he could not do it otherwise than by trial. But Bhâskara II succeeded in evolving a very simple and elegant method by means of which one can derive an auxiliary equation having the required interpolator ± 1, ± 2 or ± 4, simultaneously with its two integral roots, from another auxiliary equation empirically formed with any simple integral value of the interpolator, positive or negative. This method is called by the technical name *Cakravâla* or the "Cyclic Method."¹