Bhaskara IIBhaskara II

Casey Gregory

Background InformationBackground Information

• One of most famous Indian mathematicians• Born 1114 AD in Bijjada Bida • Father was a Brahman (Mahesvara) and astrologer• Nicknamed Bhaskaracharya “Bhaskara the

Teacher”• Studied Varahamihira and Brahmagupta at Uijain

What he knewWhat he knew

• Understood zero and negative numbers• Except how to divide by it

• Knew x^2 had 2 solutions *

• Had studied Pell’s equation and other Diophantine problems

His AccomplishmentsHis Accomplishments• First to declare a/0 = *

• First to declare + a = • Wrote 6 works including

• Lilavati (mathematics)• Bijaganita (algebra)• Siddhantasiromani• Vasanabhasya (commentary on Siddhantasiromani)• Karanakutuhala (astronomy)• Vivarana

LilavatiLilavati

O girl! out of a group of swans, 7/2 times the square root of the number are playing on the shore of a tank. The two remaining ones are playing with amorous fight, in the water. What is the total number of swans?

LilavatiLilavati13 Chapters

• definitions; arithmetical terms; interest; arithmetical and geometrical progressions; plane geometry; solid geometry; the shadow of the gnomon*; the kuttaka; combinations.

• 2 Methods for multiplication* • 4 methods for squaring• Rules of three, five, seven and nine• Kuttaka Method

• Example: “Say quickly, mathematician, what is that multiplier, by which two hundred and twenty-one being multiplied, and sixty-five added to the product, the sum divided by a hundred and ninety-five becomes exhausted.”

• Bhaskaracharya is finding integer solution to 195x = 221y + 65. • He obtains the solutions (x,y) = (6,5) or (23,20) or (40, 35) and so on.

BijaganitaBijaganita

• 12 Chapters• Including: positive and negative numbers; zero; the

unknown; surds*; the kuttaka*; indeterminate quadratic equations; simple equations; quadratic equations; equations with more than one unknown; quadratic equations with more than one unknown; operations with products of several unknowns; and the author and his work

• Quadratic equation - 700 A.D. Brahmagupta who also recognized 2 roots in the solution. 1100A.D. ANY positive number has 2 square roots

• Tried to prove a/ 0 = , however if that were true, *0 = a, therefore proving all numbers equal

• Shows that the kuttaka method to solve indeterminate equations such as ax + by + cz = d has more than one solution.

• His conclusion shows his poetic and passionate nature:• “A morsel of tuition conveys knowledge to a comprehensive

mind; and having reached it, expands of its own impulse, as oil poured upon water, as a secret entrusted to the vile, as alms bestowed upon the worthy, however little, so does knowledge infused into a wise mind spread by intrinsic force.”

Siddhanta SiromaniSiddhanta Siromani

• Picture of Goladhyaya.

Siddhanta SiromaniSiddhanta Siromani

• Wrote Siddhanta Siromani (1150 AD)• Leelavati (arithmetic)• Bijaganita (algebra)• Goladhayaya (spheres, celestial globes)• Grahaganita (mathematics of the planets)

Topics Covered in Siddhanta Siromani

Topics Covered in Siddhanta Siromani

• Astronomy Related• Latitudes & longitudes of the planets; three problems of

diurnal* rotation; syzygies*; eclipses; the moon's crescent; conjunctions of the planets with each other and stars

• Sphere Related• “nature of the sphere; cosmography and geography;

planetary mean motion; eccentric epicyclic model of the planets; the armillary sphere; spherical trigonometry; ellipse calculations; first visibilities of the planets; calculating the lunar crescent; astronomical instruments; the seasons; and problems of astronomical calculations.

Further Information in SiddhantaFurther Information in Siddhanta

• First time trigonometry was studied as it’s own entity, rather than how it related to other calculations.• sin(a + b) = sin a cos b + cos a sin b

• sin(a - b) = sin a cos b - cos a sin b.

His 7th work?His 7th work?

• There exists a 7th work, but it is thought to be a forgery.

After Bhaskara IIAfter Bhaskara II

• Bhaskara II dies in 1185• A HUGE scientific lull after invasion by

muslims• 1727, next important Hindu mathematician

Sawai Jai Singh II• Several of Bhaskara’s findings were not

explored heavily after his death, and ended up being “discovered” later by European mathematicians.

Bhaskara II RediscoveredBhaskara II Rediscovered

• chakrawal, or the cyclic method, to solve algebraic equations. *• 6 centuries later, Galois, Euler and Lagrange

rediscovered this and called it "inverse cyclic".

• differential calculus• Rediscovered as "differential coefficient" • "Rolle's theorem"• Newton and Leibniz receive credit

• Bhaskara is renowned for his concept of Tatkalikagati (instantaneous motion).

Works CitedWorks Cited

• http://www.ilovemaths.com/ind_mathe.htm

• http://www.bbc.co.uk/dna/h2g2/A2982567

• http://www-groups.dcs.st-and.ac.uk/%7Ehistory/Mathematicians/Bhaskara_II.html

• http://www.math.sfu.ca/histmath/India/12thCenturyAD/Bhaskara.html