Archimedes

Archimedes is remembered as the greatest mathematician of the ancient era. He contributed significantly in geometry regarding the areas of plane figures and the areas as well as volumes of curved surfaces. His works expected integral calculus almost 2000 years before it was invented by Sir Isaac Newton and Gottfried Wilhelm von Leibniz. He also proved that the volume of a sphere is equal to two-thirds the volume of a circumscribed cylinder. He regarded this as his most vital accomplishment. So, he desired that a cylinder circumscribing a sphere ought to be inscribed on his tomb. He found an approximate value of pi by circumscribing and inscribing a circle with regular polygons of 96 sides. His works have original ideas, impressive demonstrations and excellent computational techniques. Some of these which have survived are:

on the sphere and cylinder
measurement of a circle
on conoids and spheroids
on spirals
on plane equilibriums
the sand reckoner
quadrature of the parabola
on floating bodies
stomachion

Euclid

Euclid is the most famous mathematician of all time. "Euclid's Elements" is divided into 13 books.

the initial six are related to plane geometry
seven, eight and nine are pertaining to number theory
number ten is regarding Eudoxus's theory of irrational numbers
eleven to thirteen comprise solid geometry
the last part throws light on the properties of five regular polyhedrons and an evidence that there can be maximum five of these

These Elements have an impressive clarity regarding the selection and order of the theorems and problems. There are minimum assumptions, less extraneous material and an excellent logic in the propositions. The Elements was first published in 1482. The other works of Euclid which survive are:

optics
phaenomena
on divisions of figures
data

The works of Euclid that have not survived are:

elements of music
book of fallacies
conics
porisms
surface loci

Pythagoras

He was a Greek mathematician. His belief was that all relations could be expressed as number relations i.e. all things are numbers. He deduced this conclusion due to observations in mathematics, music and astronomy. The Pythagorean theorem is thought to be first proved by the Pythagoreans. However, it is thought that this was known in Babylonia, where Pythagoras traveled in his young days. The Pythagoreans also observed that vibrating strings created harmonious tones if the ratios of the length of the strings are whole numbers. These ratios could be extended to other devices also. The important discovery was that the diagonal of a square was not an integral multiple of its side. This led to the proof of existence of irrational numbers.

Blaise Pascal

The French mathematician had been involved in imaginative and subtle work in geometry and other branches of mathematics. In 1645, Pascal invented the first calculating machine and sold it. His work in hydrostatics led to the invention of the syringe and hydraulic press. In 1647, he published an essay on conic sections using the methods of Gerard Desargues and deserted the field of mathematics. However, later he developed an interest in probability due to his involvement in gambling.

Aryabhatta

"Aryabhatiya" is the name of Aryabhatta's work. There are an introductory 13 verses followed by 108 verses, all of them divided into 4 chapters. Aryabhatta found out the approximate value of pi and writes about it in the second part of his works (Ganitapada 10). It is possible that he found out that pi is irrational. In Ganitapada 6, he mentions the formula to calculate the value of a triangle. He developed the "Kuttaka" method to solve first order Diophantine equations. This is termed as the "Aryabhatta algorithm". The number place-value system was obviously present in his work. Later, this system was noticed in the 3rd century Bakhshali manuscript. Georges Ifrah, the French mathematician, states that the number "zero" was implicit in this system.

Some other famous mathematicians are Stefan Banach, Georg Cantor, Joseph Fourier, John von Neumann, and Brook Taylor.

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