Arithmetic - (Aryabhatta, 476 AD)
Hindu - Arabic number system is consdered to be a universal human language, without which mathematics, science and commerce would be almost impossible. Russian mathematician Michel Ostrogradski stated invention of the Hindu - Arabic number system is the greatest discovery after writing. Aryabhatta worked on different place value notations and finally developed the decimal place value notation and the place holder. This number system and mathematics went to Europe and was known as "Modus Indorum" or "method of the Indians" during the middle ages. This method of the Indians is none other than our arithmetic today.
Plain Geometry - (Euclid, 350 BC)
In geometry Euclid's "elements" is considered to be the major guide in the ancient world. Euclid's elements was translated to Arabic and then to Latin during the renaissance. Euclid's elements have many geometric relations and proof of them. Euclid started with known axioms and developed the proof using them. This method is known as axiomatic method and considered to be the crowning glory of Euclid and Greek mathematics in general. The axiomatic method is highly venerated in modern mathematics. All equations and relations in mathematics are proven today using the axiomatic system somewhat similar to Greek method. Axiomatic system of Greeks is the foundation of physices, mathematics and some fields of Engineering.
Solid Geometry - (Archimedes, 250 BC)
Archimedes took plain geometry to a higher level by considering solid geometry such as cones and spheres. Archimedes is especially important for his discovery of the relation between the surface and volume of a sphere and its circumscribing cyclinder. Archimedes also famous for his theorem on displacement of water from a solid body. Other than solid geometry, Archimedes also worked on mechanics and hydraulics. Methods used by Archimedes was used by Newton to develop calculus. Mathematics of Archimedes is used for many diverse fields ranging from Engineering, Hydraulic, Mechanics and Chemical Engineering. As per Greek historian Proclus of 4th century AD, Roman soldiers would run away when they see a rope or a lever fearing that "Archimedes has deviced another machine to kill them". This great mathematician was killed by a Roman soldier while working on a problem on the sand.
Algebra - (Brahmagupta, 596 AD)
Brahmgaupta is known as the man who found zero, most priced discovery of mathematics. He incorporated the zero and negative numbers to our number system which was developed by Aryabhatta. Brahmagupta solved many different types of equations using general methods. Prior to Brahmagupta, Greek mathematician Diophantus solved equations using special - ad hoc methods. Brahamagupta solved first order indeterminate equations, second order determinate (quadratic) and second order indeterminate equations and equations with multiple variables. Al Biruni of 10th century says that Brahmgupta's Brahmasputha Siddhanta (or a simplified version of it) was translated to Arabic as Sind Hind due to an order given by Caliph Al Mansoor in 776 AD.
It is Brahmagupta who developed rules for four fundamental operations using Indian numerals, which the world is using today. Pythogaros developed methods using Greek numbers.
Our sine and cosine functions come from Aryabhatta's Ardha Jya and Kotti Jya which were translated to Arabic as Jiba. Arab mathematicians added tangent function. Jiba was translated to Latin as Sinus. It is widely accepted that Aryabhatta obtained Ptolemy's full chord method and transformed it to much easier half chord method. Prior to that Ptolemy used chords instead of half chords. Some say Hipparchus used chords even before Ptolemy. This claim is unsubstantiated.
Newton and Leibnitz discovered calculus at the same time period. Newton used differential calculus to prove his methods while Liebniz used integral calculus. Other than calculus, both worked on other fields. Newton is specially famous for his theory on gravitation which stood for many centuries before Einstein came up with the theory of relativity.
Algebraic - Geometry (Rene Descarte):
Rene Descarte combined Algebra and Geometry. (To be contd...)
Algebraic Notation (Francoise Vieta)
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