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Contents
Newton & Leibniz
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Newton and Leibniz
Issac Newton (1642 - 1727) Issac Newton is famous for his work in mathematics and physics, especially in the fields of calculus, optics and gravitation. Although, contrary to popular thought, he did not invent calculus to aid his work in mechanics, he realised its necessity. Born into a poor family, his inclination towards mathematics was not realised until he was given the chance to study in Cambridge. He is believed to have taken an interest in mathematics after reading a book on astronomy and realising he needed trigonometry to fully understand it. It has been documented that he was inspired by Euclid and Descartes. Newton's teacher, Issac Barrow, had theories on tangents of curves, as well as finding lengths of curves and areas bounded by them. Barrow's method resembled differentiation, but he did not realise its significance. Newton used the concept of limits to expand on Barrow's ideas. These ideas led him to develop his famous theory of fluxions and fluents, known today as differential calculus. He used these ideas in describing motion. His algorithms had included the product and chain rules. Maxima and minima problems were the next problems he tackled. Many of his ideas were developed from the power series. One of which is the binomial theorem, which he found by studying patterns in power series. The theorem he developed was initially in integral form, though he eventually modified it. Newton also further expounded on Barrow's theorems for finding plane areas. He related differentiation and integration as reverse processes. He worked on a table of integrals. These ideas aided in finding such areas. He wrote about the modern rules of substitution and integration by parts. His most famous work is the Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), where he wrote about calculus and physics.
Gottfried Wilhelm Leibniz (1646 - 1716) Leibniz is the other inventor of calculus. During his time, he and Newton argued over the ownership of their discoveries, each staking a claim as the inventor of calculus. This dogfight eventually involved many prominent mathematicians all over Europe, and remains pretty much unresolved. Leibniz initially intended to study law, but was rejected by the law school. He eventually developed an interest in mathematics and was influenced by Pascal's works. The notation that he developed eventually became the standard notation we use today, mostly because it is easier to use and understand compared to Newton's. One problem he tackled was, deriving the equation of a curve from the properties of its tangents. He eventually recognised that the method is integration, and hence made the connection between differentiation and integration. The present-day notation for integration, the elongated S, was introduced by Leibniz. It stands for sum, as his integration involved sums and differences. Eventually he introduced the modern notation of
Another area he studied was whether d (xy) = dx.dy and its quotient equivalent. He proved it false, and eventually determined the product and quotient rules of differentiation. He also came up with the rule d (xn) = nxn-1 (he used d to represent differentiation, unlike our modern notation of d/dx). The formula for the surface of revolution of rotating a curve about the x-axis was also his accomplishment. He tackled the maxima-minima problem, as well as investigated differential equations.
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