Bhaskara II
Casey Gregory
Background Information
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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 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 Accomplishments
• First to declare a/0 = *
• First to declare + a = 
• Wrote 6 works including
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Lilavati (mathematics)
Bijaganita (algebra)
Siddhantasiromani
Vasanabhasya (commentary on Siddhantasiromani)
Karanakutuhala (astronomy)
Vivarana
Lilavati
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?
Lilavati
13 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.
Bijaganita
• 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 Siromani
• Picture of Goladhyaya.
Siddhanta Siromani
• Wrote Siddhanta Siromani (1150 AD)
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Leelavati (arithmetic)
Bijaganita (algebra)
Goladhayaya (spheres, celestial globes)
Grahaganita (mathematics of the planets)
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 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?
• There exists a 7th work, but it is thought to
be a forgery.
After 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 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 Cited
• http://www.ilovemaths.com/ind_mathe.htm
• http://www.bbc.co.uk/dna/h2g2/A2982567
• http://www-groups.dcs.stand.ac.uk/%7Ehistory/Mathematicians/Bhas
kara_II.html
• http://www.math.sfu.ca/histmath/India/1
2thCenturyAD/Bhaskara.html