Part 24 - Kalasavarna (reduction to a simple fraction) and Vallisavarnana

KALASAVARNA It seems that owing to the lack of proper symbols required for indicating mathematical operations, the Indian Mathematicians have divided kalasavarna (reduction to a simple fraction) into various classes known as jatis. For instance, Skandasena, Sripati and Bhaskara II have given four jatis, whereas Brahmagupta has given five, and Sridhara and Mahaviracarya each, six. The four jatis above referred to are:- 2 e (1) Bhaga¹i. e. the form (1.) (2) Prabhaga i. e. the form (off of 3 of 3) of ...) F (3) Bhaganubandha or amsanubandhas. This is of two types :- (i) the form (a+b)*%; and (ii) the form + of b d a + of 5 + d (4) Bhagapavaha or bhagapavahana. This is also of two types:- (i) the form (a - b); 06 06 and (ii) -2 of 3 of (1 - 1 of 7). 5 d b 1 This is translated as a 'simple fraction' by M. Rangacarya on p. 297 (Ganitasarasangraha). On this page we find bhagabhaga, bhaganubandha and bhagapavaha translated as 'a complex fraction', 'fractions in association' and 'dissociated fractions' respectively. 2 For a negative sign, a dot is usually and slightly over it. Cf. "a placed to the right of the number aa"-Bijaganita (p. 2) of Bhaskara II. In a Bakhshala Ms. recently acquired by Dr. Datta, the simple cross (+) is used for the same purpose. This is what he writes to me. 3 This is the term used by Sripati. See p. 34. 4 This form represents an association (anubandha) of an integer (rupa) and a part i. e. to say a fraction (bhaya). So it is styled as rupa-bhaganubandha or bhaganubandhasahitanka. See p. 35. 5 This form represents an association of bhaga of bhaga. So it goes by the name of bhaga-bhaganubandha or bhaganubandhabhaga. See p. 35. 6 Practically there is no difference between this and bhaganubandha except that herein there is subtraction or decrease, instead of addition or increase as is the case with bhaganubandha. VALLISAVARNANA As the next topic Sripati introduces on p. 39, vallisavarnana, and on the same page gives us a rule whereby we can reduce a chain of measures into a proper fraction.

Over and above these four jatis³, some enumerate two more. They are:- (1) Bhaga-bhaga i. e. the form (a / 2) 32 or ((( + - ). / (2) Bhaga-matr i. e. combinations of two or more of the five jatis mentioned above.³

l. c., p. 190 H. 2 Since there seems to be no special symbol to indicate division, this is written as 04 b b and it is hence likely that this may be confounded with a +C also written in the same way. 3 Mahaviracarya has noted that there can be 26 (=5 C 2+503 +504 +505) combinations.