Archimedes (287 BCE – 212 BCE) was one of the greatest mathematicians of antiquity. He was born in Syracuse, Italy and probably studied in Alexandria under Euclid. Archimedes made fundamental contributions to statics, hydrostatics, and engineering. He is credited with inventing a screw mechanism to raise water and discovered the principle of buoyancy while taking a bath. Archimedes also made advances in geometry and is believed to have anticipated the development of calculus. Much of what is known about his life comes from writings after his death.

2. Archimedes (287 BCE – 212 BCE), the greatest
mathematician and mathematical physicist of
antiquity, was born in Syracuse in what is now Italy.
His father Phidias was an astronomer.
Archimedes probably studied in Alexandria with the
followers of Euclid. Archimedes was the founder of
the sciences of statics and hydrostatics, as well as
being an ingenious engineer who used his talents to
solve a wide range of practical problems. He also
developed the principle of the lever and of multiple
pulleys. He is credited with inventing a screw
mechanism used to raise water from lower levels, but
the Egyptians may have known of the Archimedean
screw much earlier. His mathematical proofs were
both boldly original and possessed rigor matching the
highest standards of contemporary geometry. Much
of what is known of his life is anecdotal, coming from
the writings of historian, biographer and philosopher
Plutarch, born 250 years after Archimedes’ death.

3. 1. Discovered how to find the volume of a sphere and
determined the exact value of pi .
2.Principle of Buoyancy. (It is believe that when he
discovered the principle of Buoyancy, he went running
through the streets naked shouting 'Eureka' - I have
found it)
3.It is believed that he was actually the first to have
invented integral calculus, 2000 years before Newton
and Leibniz.
4.Power of Ten a way of counting that refers to the
number of 0's in a number which eliminated the use of
the Greek alphabet in the counting system. (Scientific
Notation)
5.A formula to find the area under a curve, the amount of
space that is enclosed by a curve.

4. The treatise On plane equilibriums sets out the fundamental
principles of mechanics, using the methods of geometry. Archimedes
discovered fundamental theorems concerning the centre of gravity of
plane figures and these are given in this work. In particular he finds,
in book 1, the centre of gravity of a parallelogram, a triangle, and a
trapezium. Book two is devoted entirely to finding the centre of
gravity of a segment of a parabola. In the Quadrature of the parabola
Archimedes finds the area of a segment of a parabola cut off by any
chord.
The first book is in fifteen propositions with seven , while the
second book is in ten propositions. In this work Archimedes explains
the Law of Lever, stating, "Magnitudes are in equilibrium at distances
reciprocally proportional to their weights."
Archimedes uses the principles derived to calculate the areas and
of various geometric figures including triangles, Parallelogram and
parabolas.
Archimedes once said :
Give me a place to stand on, and I will move the
Earth.

5. On floating bodies is a work in which
Archimedes lays down the basic principles
of hydrostatics. His most famous theorem
which gives the weight of a body immersed
in a liquid, called Archimedes' principle, is
contained in this work. He also studied the
stability of various floating bodies of
different shapes and different specific
gravities.

6. In Measurement of the Circle Archimedes shows that
the exact value of π lies between the values 310/71 and
31/7. This he obtained by circumscribing and inscribing a
circle with regular polygons having 96 sides.
This is a short work consisting of three propositions. It is
written in the form of a correspondence with Dositheus
of Pelusium, who was a student of . In Proposition II,
Archimedes shows that the value of (pi) is greater than
223⁄71 and less than 22⁄7. The latter figure was used
as an approximation of π throughout the Middle Ages and
is still used today when only a rough figure is required.

7. Archimedes’ Principle : When a
body is immersed in a fluid, it
experiences an upward force equal
to the weight of the displaced
fluid.

8. The king of Syracuse, Hiero, wanted a golden
crown. He weighed a lump of gold and ordered a
goldsmith to make him a crown with it. The
goldsmith returned with a crown that weighed
exactly the same as the lump of gold. The king
was happy.
After a while, however, the king grew suspicious.
The goldsmith might not have put all the gold
into the crown. He might have used some other
metal. The king called Archimedes to help him
determine whether any other metal had been
mixed into the crown.
Archimedes thought and thought, but he couldn't figure out a way to determine
whether the goldsmith had melted silver in the crown. One day he decided to go
to the baths. As he got into the tub, some water sloshed out. He had solved his
problem. Archimedes ran out on the streets shouting "Eureka, Eureka!" He was
so excited that he hadn’t bothered to put his clothes on!
The experiment was very simple. Archimedes filled a jar to the brim, dropped
the crown in it, and gathered the water that flowed out. Then he replaced the
water in the jar and dropped in a lump of gold the same weight as the crown. He
found that the lump of gold caused less water to overflow than the crown. This
meant that the crown occupied more volume, or space, than the lump of gold.
This, in turn meant that the crown was not all-pure gold and that the king had
been cheated.

9. Euclid is one of the world's most
famous mathematicians, yet very little
is known of his life, except that he
taught at Ptolemy’s university at
Alexandria, Egypt. Euclid's Elements, a
work on elementary geometry and other
topics, exceeded other works of its
time, which are now known only by
indirect reference. The Elements
begins with definitions, postulates, and
axioms, including the famous fifth, or
parallel, postulate that one and only one
line can be drawn through a point
parallel to a given line. Euclid's decision
to make this postulate not
demonstrable assumption led to
Euclidean geometry. It was not until
the 19th century that the fifth
postulate was modified in order to
develop non-Euclidean geometry

10. The Elements are divided into 13 books.
The first 6 are on geometry; 7, 8 and 9 are
on number theory; and book number 10 is
on Eudoxus's theory of irrational numbers.
Books 11, 12, and 13 all concern solid
geometry, and end with a discussion of the
properties of the five regular polyhedrons
and proof that there can only be these
five. Euclid's Elements are remarkable for
the clarity with which the theorems and
problems are selected and ordered. The
propositions proceed logically and
concisely, with very few assumptions.

11.  The most celebrated mathematician
during the classic period. Many honors
have been placed on him and also he is
the birthplace of many mathematical
theorems functions etc.
 He was born in 476 AD with many
controversy over his birth place; some
say he was born in Kodungallor,Kerala
some atribute it to Taregna,Bihar.
 He is known for his famous treatise
Aryabhatiya written in 499 AD when he
was 23.
 Credits confered to him include value
of pi, earth’s rotation time
period,extraction of cube root of a
number, indeterminate equations,

12.  Aryabhata worked on the approximation for pi , and may
have come to the conclusion that is irrational.
 caturadhikam satamastagunam
dvasastistathasahasranam
ayutadvayaviskambhasyasanno vrttaparinahah.
"Add four to 100, multiply by eight, and then add
62,000. By this rule the circumference of a circle with
a diameter of 20,000 can be approached."
 This implies that the ratio of the circumference to the
diameter is ((4 + 100) × 8 + 62000)/20000
= 62832/20000 = 3.1416, which is accurate to
five significant figures.
 After Aryabhatiya was translated into Arabic this
approximation was mentioned in Al-Khwarizmi's book on
algebra.

13.  Aryabhata gives the area of a triangle as
tribhujasya phalashariram samadalakoti bhujardhasamvargahthat
translates to: "for a triangle, the reult of a perpendicular with the
half-side is the area”.
 Aryabhata discussed the concept of sine in his work by the name
of ardha-jya, which literally means "half-chord”.
 Today known as diophantine equation; the indeterminate equation
was always discussed in Aryabhatiya .
 His method of soving was as follows
N=ax+c=by=d or ax-by=k
 His contribution to algebra was also sophisticatedly important.

14.  He was born at Bori, in Parbhani district of
Maharashtra state in India in 7th century.
 He was the first to write Hindu-Arabic numerals and
with zero with a circle.
 He was an exponent of Aryabhatta, named
Aryabhatiyabhasya.
 He gave importance to sine function in
Aryabhatiyabhasya.
 He represented number using nonliving and living thing
For eg:- 1 was for moon , 2 was for eyes,wings etc, 5 was
for the senses of humans.
 His famous books include Laghubhaskariya and
Mahabhaskariya.
 Not to be mistaken for Bhaskara II of 12th centuary

15.  He was an astrologer manly but was also a mathematician.
 He was born in 6th centuary in Ujjain and considered to be one
of the nine jems of Vikramaditya II.
 The trigonometric formulas:
 His famous work is Panchasidanthika.

16.  Bramagupta belonged to the city of ujjain .
 Regarded as the man who used zero as a number,
negative numbers.
 The statement a negative integer multiplied by a
negative integer give a positive integer and many other
fundamental operation first appeared in his treatise
Bhramasphutasiddhanta. But how he came to the
conclusion was unknown.
 He gave basic idea to the d-quadratic method of solving.
 The following identity was attributed to him
x2 - y2 = (x + y)(x - y)
 He gave some information about what is today known as
Pythagorean triplets.
 He is also known to give a touch to today’s Pell equation

17. BRAHMAGUPTA’S
FORMULA :

18.  He was a jain Mathematician
 His celebrated work was Ganithasarangraha.
 He showed ability in quadratic equations, indeterminate
equations.