Indian Mathematics c6.16

c6.16

Āryabhaṭa’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 Āryabhaṭa is the one which contains an extremely accurate value of the ratio of the circumference of a circle to the diameter, *viz.*, π = 3 177/1250 (= 3·1416); but it is rather extraordinary that Āryabhaṭa himself never utilised this value, that it was not used by any other Indian mathematician before the 12th century and that no Indian writer quotes Āryabhaṭa as recording this value. Other noteworthy points are the rules relating to volumes of solids which contain some remarkable inaccuracies, *e.g.*, the volume of a pyramid is given as *half* the product of the height and the base; the volume of a sphere is stated to be the product of the area of a circle (of the same radius as the sphere) and the root of this area, or π3/2 r³. Similar errors were not uncommon in later Indian works. The rule known as the *epanthem* occurs in Āryabhaṭa's work and there is a type of definition that occurs in no other Indian work, *e.g.*, "The product of three equal numbers is a cube and it also has twelve edges."