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05 Development of Calculus

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A Paper on Indian Maths.
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    KERALA SCHOOL OF ASTRONOMY AND DEVELOPMENT OF CALCULUS M.D.SRINIVAS CENTRE FOR POLICY STUDIES mdsrinivas50@gmail.com  BACKGROUND TO THE DEVELOPMENT OF CALCULUS (c.500-1350) Zero, infinitesimals and infinity ã   Background 1)   The concept of  p ū rõa  - Śā ntimantra  of  Īś  opaniùad  . 2)   The concept of lopa  in P ā õini, abh ā va  in Ny ā ya and śū nya  in Bauddha philosophy. 3)   Śū nya  as symbol in Chandaþ S  ū tra  (VIII.29) of Piïgala (c.300BC) ã   Brahmagupta (c.628) on the mathematics of zero. The notion of taccheda . ã   Bh ā skara II (c.1150) on the notion of khahara  or ananta    Zero, infinitesimals and infinity Bh ā skar ā c ā rya, while discussing the mathematics of zero in  L ī  l ā vat  ī  , notes that when further operations are contemplated, the quantity being multiplied by zero should not be changed to zero, but kept as is. Further, when the quantity which is multiplied by zero is also divided by zero, then it remains unchanged. He follows this up with an example and declares that this kind of calculation has great relevance in astronomy. What is the number which when multiplied by zero, being added to half of itself multiplied by three and divided by zero, amounts to sixty-three?   [लीलावती ४४-४६ वासना]    Zero, infinitesimals and infinity   Bh ā skara, it seems, had not fully mastered this kind of calculation with infinitesimals as is clear from some of the examples he considers in  B ī   jagaõita , while solving quadratic equations by eliminating the middle term ( ekavarõa-madhyam ā haraõa ).  कः वाध सिहतो रािशः खगु णो वगतो यु तः ।  वपदायां खभ जाताः पदशोयताम् ।। [बीजगिणतम् १३६]   [{0 (  x + (  x/2 )} 2  + 2 {0(  x + (  x/2 ))}]/ 0 = 15. Bh ā skara in his V  ā san ā  just cancels out the zeroes and obtains  x = 2 .  

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Jul 23, 2017

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