MATHEMATICIANS AND THEIR ENORMOUSINVENTIONS

ARCHIMEDES 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.

CONTRIBUTION1.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.

ON PLANE EQUILIBRIUMS

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.

ON FLOATING BODIES

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.

MEASUREMENT OF THE CIRCLE

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.

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

THE GOLD CROWN 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.

EUCLID 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

CONTRIBUTION 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.

ARYABHATTA 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, trigonometry etc.

ARYABHATTA AND PI Aryabhata worked on the approximation for pi , and

may have come to the conclusion that  is irrational. caturadhikam satamastagunam

dvasastistathasahasranamayutadvayaviskambhasyasanno 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.

ARYABHATTA AND TRIGONOMETRY, INDETERMINATE EQUATION AND ALGEBRA  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.

BHASKARA 1 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

thingFor 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

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.

VARAHAMIHIRA

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

BRAHMAGUPTA

BRAHMAGUPTA AND CYCLIC QUADRILATERAL

BRAHMAGUPTA’S FORMULA :

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

indeterminate equations.

MAHAVIRA

MADE BY : APURVA

PARASHAR & PRABHAT GAURAV

CLASS : X ‘A’

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