Sridharacharya’s contribution to Geometry

RADHIKA YELKAWARASSOCIATE PROFESSORDEPARTMENT OF MATHEMATICSL.A.D. COLLEGENAGPUR

SRIDHARACHARYA (c. 870– c. 930)

Sridharacharya is believed to have lived between seventh to eleventh century.

The best present estimate is 900 ADBirth place- Hooghly District in West

Bengal or South IndiaFather's name - Baladevacharya Mother's name- Acchoka.

Works Sridhara is known as 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).

Works (contd……)

Of all 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.

Patiganita

Patiganita is the most 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

Patiganita

First Part• money measure• weights• measure of capacity• linear measure • time measurement

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.

Patiganita Third part

simple interest 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.

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

Trisatika

This work of 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.

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 ( ) between circumference of a circle and its 𝝅diameter.

The value of

Formula for volume of the sphere i.e. was first derived by Archimedes. This formula uses a symbol . 𝜋

is an irrational number and therefore there is no exact value of 𝜋. The value of now we know is 3.14

V

= 4.19-----(1)

Volume of sphere Archimedes formula

Meaning:Half the cube of 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

+()

where d = 2r, r = radius of a sphere

-----------(2)

Volume of sphere

Value of

Thus the volume of sphere computed by Sridharacharya is very close to

the value computed by Archimedes and it differs only with the value of .

From (1) and (2)

Thus It appears that Sridharacharya considered the value of

Area and Circumference of a circle

Sridhara has computed value of 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 diameteri.e. Circumference of circle = =

Meaning of second line- The area of a circle is the square root of the product of 10 with the square of semi-diameter squareArea of circle = =

Area of segment of 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 =

=