He was renowned as an astronomer and as such tried to introduce sounder views of that science but was bitterly opposed by the orthodox. The mathematical work attributed to him consists of thirty-three couplets into which is condensed a good deal of matter. Starting with the orders of numerals he proceeds to evolution and involution, and areas and volumes. Next comes a semi-astronomical section in which he deals with the circle, shadow problems, etc.; then a set of propositions on progressions followed by some simple algebraic identities. The remaining rules may be termed practical applications with the exception of the very last which relates to indeterminate equations of the first degree. Neither demonstrations nor examples are given, the whole text consisting of sixty-six lines of bare rules so condensed that it is often difficult to interpret their meaning. As a mathematical treatise it is of interest chiefly because it is some record of the state of knowledge at a critical period in the intellectual history of the civilised world; because, as far as we know, it is the earliest Hindu work on pure mathematics; and because it forms a sort of introduction to the school of Indian mathematicians that flourished in succeeding centuries.
Āryabhata's work contains one of the earliest records known to us of an attempt at a general solution of indeterminates of the first degree by the continued fraction process. The rule, as given in the text, is hardly coherent but there is no doubt as to its general aim. It may be considered as forming an introduction to the somewhat marvellous development of this branch of mathematics that we find recorded in the works of Brahmagupta and Bhāskara. Another noteworthy rule given by Āryabhata is the one which contains an extremely accurate value of the ratio of the circumference of a circle to the diameter, viz., ; but it is rather extraordinary that Āryabhata himself never utilised this value, that it was not used by any other Indian mathe-
