Euclid

                                                  



Euclid (sometimes called Euclid of Alexandria), was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I (323 - 283 BC). His elements is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics especially geometry from the time of its publication until the late 19th or early 20th century. In the elements, Euclid deducted the principles of what is now called Euclidean Geometry from a small set of axioms. Euclid also wrote on  perspective, conic section, spherical geometry, number theory and rigor.

Contribution of Euclid in mathematics:

1. Euclidian geometry states that sum of the angles of a triangle is 180 degree.
2. Euclid formulated a method to find out the H.C.F.
3. He gave a proof that prime numbers are infinite.
4. It was Euclid who first proved root of 2 as a transcendental number.

Pythagoras

Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and has been credited as the founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived so very little reliable information is known about him. He was born on the island of Samos, and traveled, visited Egypt and Greece, and maybe India, and in 520 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild.

Pythagoras made influential contribution to philosophy and religion in the late 6th century BC. He is often reverted as a great mathematician and scientist and is best known for the Pythagorean theorem which bears his name.Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy.

Contribution of Pythagoras in Mathematics:

1. The sum of the angles of a triangle is equal to two right angles.
2. The theorem of Pythagoras for a right angles triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. The Babylonians understood this 1000 years earlier, but Pythagoras proved it.
3. The constructing figures of a given area and geometrical algebra. For example they solved various equations by geometrical means.
4. The discovery of irrational numbers is attributed to the Pythagoreans, but seems unlikely to have been the idea of Pythagoras because it does not align with his philosophy the all things are numbers, since number to him meant the ratio of two whole numbers.
5. The five regular solids (tetrahedron, cube, octahedron, icosahedron, dodecahedron). It is believed that Pythagoras knew how to construct the first three but not last two.
6. Pythagoras taught that Earth was a sphere in the center of the universe, that the planets, stars, and the universe were spherical because the sphere was the most perfect solid figure. He also taught that the paths of the planets were circular. Pythagoras recognized that the morning star was the same as the evening star, Venus

Bhaskaracharya

Bhaskara (also known as Bhaskaracharya), (1114 - 1185), was an Indian mathematician and astronomer. He was born in Bijapur in modern Karnataka.

Bhaskara and his works represents a significant contribution to mathematical and astronomical knowledge in 12th century. He has been called the greatest mathematician of medieval India. His main work SIDDHANTHA SHIROMANI, (Sanskrit of "Crown of Treaties") is divided into  four parts called LILAVATI, BIJAGANITA, GRAHAGANITA and GOLADHYAYA, which are also sometimes considered four independent works. These four section deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treaties named KARANA KAUTUHALA.

Bhaskara's work on calculus predates Newton and Leibniz by over half a millennium. Hes is particularly known in the discovery of the principles of differential calculus and its application to astronomical prob;ems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhaskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus.



Some of Bhaskara's contributions to mathematics include the following:

• A proof of the Pythagorean Theorem by calculating the same area in two different ways and then canceling out terms to get a + b = c.

• In Lilavati, solutions of quadratic and cubic indeterminate equation are explained.

• Solutions of indeterminate quadratic equations (of the type ax + b = y).

• Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century

• A cyclic Chakravala method for solving indeterminate equations of the form ax + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.

• The first general method for finding the solutions of the problem x − ny = 1 (so-called “Pell’s equation “)was given by Bhaskara II.

• Solutions of Diophantine Equations of the second order, such as 61x + 1 = y. This very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat , but its solution was unknown in Europe until the time of Euler in the 18th century.

• Solved quadratic equations with more than one unknown, and found negative and irrational i solutions.

• Preliminary concept of mathematical analysis.

• Preliminary concept of infinitesimal Calculus, along with notable contributions towards integral calculus .
• Conceived differential calculus, after discovering the derivative and differential coefficient.

• Stated Roll’s theorem, a special case of one of the most important theorems in analysis, the mean value theorem. Traces of the general mean value theorem are also found in his works.

• Calculated the derivatives of trigonometric functions and formulae.

• In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results.