Sridharacharya[1]

Sridharacharya[1]

• 1.
  Sridharacharya’s contribution to Geometry RADHIKAYELKAWAR ASSOCIATE PROFESSOR DEPARTMENT OF MATHEMATICS L.A.D. COLLEGE NAGPUR
• 2.
  SRIDHARACHARYA (c. 870– c.930) Sridharacharya is believed to have lived between seventh to eleventh century. The best present estimate is 900 AD Birth place- Hooghly District in West Bengal or South India Father's name - Baladevacharya Mother's name- Acchoka.
• 3.
  Works Sridhara is knownas the author of two mathematical treatises, namely Patiganita Trisatika However at least three other works have been attributed to him, namely Bijaganita Navasati (having nine hundred) Brhatpati (bigger pati) Information about these books was given in the works of Bhaskara II (around 1150), Makkibhatta (in 1377) and Raghavabhatta (in 1493).
• 4.
  Works (contd……)  Ofall the Hindu Acharyas, the description of Sridharacharya on zero is the most explicit. He has written,  If 0 (zero) is added to any number, the sum is the same number.  If 0 (zero) is subtracted from any number, the number remains unchanged.  If 0 (zero) is multiplied by any number, the product is 0(zero).  He has said nothing about division of any number by 0(zero).  In the case of dividing a fraction, he has found out the method of multiplying the fraction by the reciprocal of the divisor  Wrote on practical applications of algebra and also separated algebra from arithmetic’s  One of the first to give a formula for solving quadratic equations.
• 5.
  Patiganita Patiganita is themost important work of Sridhara. Throughout the book, Sridhara has given methods to solve problems in terse rules in the form of verses No proofs are given..  The book is divided into four parts  It contains series of problems, some of which are only approximate.  Only one copy of this book is survived and its last pages are missing
• 6.
  Patiganita  First Part •money measure • weights • measure of capacity • linear measure • time measurement
• 7.
  Patiganita  Second part(Prakarama) • Addition, subtraction, multiplication, division, • squaring and square root, • cubing and cube root • operations for fractions, reduction of fractions • rule of three, inverse rule of three, rules of five, seven and nine • barter of commodities, sale of living beings etc.
• 8.
  Patiganita Third part  simpleinterest  valuation of pieces of gold  partnership  purchase and sale  wages and payments, wages paid from the commodity  series in arithmetic progression, series in geometric progressions  miscellaneous problems on series in arithmetic progression  series of squares and cubes of the terms of an arithmetic series.
• 9.
  Patiganita  Fourth part Area of quadrilateral with equal and unequal altitudes  Area of the triangle  Area and circumference of circle  Area of segment of circle  Surface area and volume of a sphere
• 10.
  Trisatika  This workof Sridhara is also known as Patiganitasara  Patiganitasara is a summary of the Patiganita including the missing portion.  It is called Trisatika as it contains 300 verses.
• 11.
  Volume of sphere Construction of perfect domes in the ancient structures is a testimony of knowledge of our ancient civilizations that there is a fixed relationship (𝝅 ) between circumference of a circle and its diameter.  The value of 𝜋 = 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒  Formula for volume of the sphere i.e. 𝑉 = 4 3 𝜋𝑟3 was first derived by Archimedes. This formula uses a symbol 𝜋.  𝜋 is an irrational number and therefore there is no exact value of 𝜋. The value of 𝜋 now we know is ≈ 22 7 ≈3.14
• 12.
  V = 𝟒𝛑 𝟑 𝐫 𝟑 = 𝟒 𝟑 × 𝟐𝟐 𝟕 𝑟3 = 88 21 𝑟3 ≈4.19 𝑟3 -----(1) Volume of sphere Archimedes formula
• 13.
  Meaning:Half the cubeof the diameter of a sphere combined with its 18th part is the volume of a sphere i.e. If d is the diameter of the sphere then 𝐕 = 𝐝 𝟑 𝟐 + 𝟏 𝟏𝟖 ( 𝐝 𝟑 𝟐 ) = 19 36 𝑑3 = 19 36 (2r)3 where d = 2r, r = radius of a sphereType equation here. = 38 9 𝑟3 ≈ (𝟒. 𝟐𝟐)𝐫 𝟑 -----------(2) Volume of sphere
• 14.
  Value of 𝜋 Thusthe volume of sphere computed by Sridharacharya is very close to the value computed by Archimedes and it differs only with the value of 𝝅. From (1) and (2) 38 9 𝑟3 = 4𝜋 3 𝑟3  38 9 = 4𝜋 3 𝜋 = 38 9 × 3 4  𝜋 = 19 6 ≅ √10 Thus It appears that Sridharacharya considered the value of 𝝅 = √𝟏𝟎
• 15.
  Area and Circumferenceof a circle Sridhara has computed value of 𝜋 in the following verse Meaning of first line- The circumference of a circle is equal to the square root of 10 times the square of its diameter i.e. Circumference of circle = 10𝑑2 = 10 × 𝑑 Meaning of second line- The area of a circle is the square root of the product of 10 with the square of semi-diameter square Area of circle = 10 𝑑 2 2 2 = 10 × 𝑑 2 2
• 16.
  Area of segmentof a circle Meaning : The square of the arrow as multiplied by half the sum of the chord and the arrow should be multiplied by 10 and divided by 9. The square root of the quotient gives the area of the segment of a circle Area of a segment of a circle = ℎ × 𝑐+ℎ 2 2 × 10 9 = 𝜋 3 ℎ × 𝑐+ℎ 2 2