The method of integral solution of indeterminate equation of the type: by=ax±c in ancient and medieval India

Abstract

Aryabhata I (476 A.D.) and other Indian scholars have given a general method of integral solution of indeterminate equations of the type : by=ax e, which involves the knowledge of continued fraction : Pn.qn-1-Qn.Pn-1= 1 according as n is even or odd, where and are the n-th and (n-1)-th successive approximations of a/b. The paper presents a detailed discussin of the knowledge used by Aryabhata I in his method of solution and of contributions made by the scholars like Bhaskara I, Aryabhata II, Bhaskara II of the rule of Aryabhata I. The original Sanskrit passages from the text of Karanapaddhati (1596 A.D.?) with English translations have also been presented in the paper to show to what extent the perfection of the technique has been attained by the author of Karanapaddhati following traditional lines and the knowledge found its way to other cultural areas.