Bramhagupta dec[1]

1. 1 May 2013 1 Presentation by Aruna Shobhane Dinanath Jr. College, Nagpur Bramahagupta

2. Born in 598 AD in Bhinmal ( Rajasthan ) In the reign of Vaghra mukha , a great king of capa dynesty , when 500 years of Saka era had elapsed , Bramhagupta son of Jishnu at the age of 30 , composed Bramhasphutsiddhanta for the pleasure of good mathematicians and astronomers. 1 May 2013 2 Ujjain School of Hindu Mathematics Authority on Astronomy and Astrology Head of the astronomical observatory at Ujjain

3. Texts on mathematics and astronomy: Brahmasphutasiddhanta in 628 A. D. (The Opening of the Universe) Khandakhadyaka in 665 A. D. (literally meaning sweet toast ) Uttar Khandakhadyaka in 672 A. D (literally meaning more sweet toast ) . Works of Bramhagupta 1 May 2013 3

4. Brahmasphutasiddhanta The Opening of the Universe Contains 24 chapters and 1080 verese . First 10 - Astronomy Solar and lunar eclipses planetary conjunctions positions of the planets Motion of celestial bodies and their speed Prediction of full moon day and New moon day Chapter 11 – Dushana Criticism of different results by previous m athematicians. 4

5. 1 May 2013 5 Chapter 12 – Ganita Study of Arithmetic, Algebra, Progressions , Permutations , Polynomials, study of Plane figures Chapter 18 – Kuttaka or Kuttakadhyya Various forms of Quiz, Puzzels . Chapter 19 to 24 – Solid Figures – solid Geometry Volume ,Surface Area – Cone Cylinder Other Properties.

6. 1 May 2013 6 Kahandakhadyaka - 5 Chapters Effects of Movements of Celestial bodies on Human life Results useful in everyday life Favorable timings of marriages , Birth Popular in Short time. UttarKahandakhadyaka – 5 Chapters Different methods of proofs – some questions posed in Kahandakhadyak

7. Telegraphic rule for Sine function BSS XXI -19 1 May 2013 7

8. The square root of the product of four factors formed by the semi-perimeter which is diminished by each side is the exact area of cyclic quadrilateral . Area Theorem Where p, q, r, s are the sides of the cyclic quadrilateral. And S , the semi perimeter, given by 1 May 2013 8

9. 1 May 2013 9 But since ABCD is a Cyclic Quadrilateral Trigonometric Proof

10. 1 May 2013 10 By law of cosines

11. 1 May 2013 11 A B C D Heron’s Formula for Triangle with sides p, q, r Bramhagupta’s Area Theorem

12. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral So given the lengths p , q , r and s of a cyclic quadrilateral, the approximate area is 1 May 2013 12

13. The sums of the products of the sides about the diagonals bee both divided by each other. Multiply [the quotients obtained] by the sum of the products of the opposite sides the square roots (of the result )are diagonals. In a cyclic quadrilateral ABCD and sides a = AB , b = BC , c = CD , and d = DA , the lengths of the diagonals p = AC and q = BD,then Bramhaguptas expression for diagonals 1 May 2013 13

14. Brahmaguptan quadrilateral Formation of cyclic quadrilateral with integer sides, integer diagonals, and integer area. The kotis and the bhujas of two Jatyas multiplied by each others hypotenuse are the four sides in a Visama Quadrilateral. If e, f, g and p, q, r are the sides (integral or rational ) of Jatyas i . e. Right Angled Triangles with g and r being hypotenuse , then e.r , f.r , g.p g.q are the required sides of Brahmaguptan quadrilateral. 1 May 2013 14

15. Construction of Brahmaguptan quadrilateral 1 May 2013 15 (e, f, g ) x p ….(Triangle with sides p.e , p.f , p.g ) (e, f, g) x q …. ( Triangle with sides e.q , f.q , g.q ) (p, q, r) x e …. ( Triangle with sides p.e , q.e , r.e ) (p, q, r ) x f …. ( Triangle with sides p.f , q.f , r.f )

16. 1 May 2013 16 Brahmaguptan quadrilateral

17. 1 May 2013 17 This is a note worthy contribution of Brahmagupta to pure mathematics

18. All Brahmagupta quadrilaterals with sides a, b, c, d , diagonals e, f , area K , and circumradius R can be obtained by the following expressions involving rational parameters t , u , and v : 1 May 2013 18

19. Brahmagupta dedicated a substantial portion of his work to geometry and trigonometry. He established √10 (3.162277) as a good practical approximation for π (3.141593) 1 May 2013 19

20. Brahmagupta's theorem Brahmagupta's theorem states that BM = MC . If a cyclic quadrilateral is orthodiagonal then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. 1 May 2013 20

21. We need to prove that AF = FD =FM To prove that AF = FM , first note that the angles FAM and CBM are equal, because they are inscribed angles that intercept the same arc of the circle. Furthermore, the angles CBM and CME are both complementary to angle BCM (i.e., they add up to 90°), and are therefore equal. Finally, the angles CME and FMA are the same. Hence, AFM is an isosceles triangle, and thus the sides AF and FM are equal. The proof that FD = FM goes similarly: the angles FDM , BCM , BME and DMF are all equal, so DFM is an isosceles triangle, so FD = FM . It follows that AF = FD , 1 May 2013 21

22. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. "non-unequal" cyclic quadrilateral (an isosceles trapezoid), the length of each diagonal is . The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes 1 May 2013 22

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