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Newton never married, but he brought his niece to London as his hostess and lived in upper class style.  

Newton never married, but he brought his niece to London as his hostess and lived in upper class style.  

==Year of great discovery==

==Year of great discovery==

The year 1666 is known as Newton's ''annus mirabilis'' (miraculous year--more precisely the two years 1665-1666), about twenty-four years of age. He later recalled, "For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more than at any time since." (By "philosophy" he meant physics.)

The year 1666 is known as Newton's ''annus mirabilis'' (miraculous year--more precisely the two years 1665-1666), about twenty-four years of age. He later recalled, "For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more than at any time since." (By "philosophy" he meant physics.)

 

===Calculus===

===Calculus===

Critical to his work on calculus was discovering the binomial theorem which expresses areas under curves as infinite series; when the curve is expressed as algebraic formula, he had invented integration. He followed that with a method for differentiation, and came upon the fundamental theorem of the calculus, that of the inverse relation of differentiation and integration. Having invented the calculus, he put aside mathematics for two years and turned to physics.

Newton broke ground with his innovative work founding the field of [[calculus]]. He had been motivated by the need for alternate ways to compute [[pi]]. He isolated a formulation of pi as the area under an [[arc]] of the [[unit circle]]; thus to calculate pi he would only have to compute this area. Whereas [[Pierre de Fermat]] had already worked out how to compute the areas under [[polynomial]] [[curve]]s, Newton faced a curve given by a formula involving a [[square root]]. To solve this problem, he re-expressed the square root in terms of an infinite sum of polynomials--this was the motivating idea for his generalized binomial theorem. The standard [[binomial theorem]] gave an expansion for ''(x+y)ⁿ'' for any nonnegative [[integer]] ''n''. The resulting expression involves binomial [[coefficient]]s. Newton's work extended this theorem to all [[real]] values of ''n'', by using [[convergent]] [[infinite series]] and generalized binomial coefficients. Therefore, to compute the area under the arc, he simply had to use Fermat's theorem to compute the area under each of the polynomial terms of the infinite series and then add them together (proving along the way that this sum converges).

 

Proceeding from this method, Newton formulated the idea of [[integration]] -- a computation of the area under any curve by using infinite series of areas. He followed that with a method for [[differentiation]], and came upon the [[fundamental theorem of calculus]], which relates differentiation and integration. Having invented the calculus, he put aside mathematics for two years and turned to physics.

 

Although Newton had communicated his discoveries in the calculus privately, he did not publish anything formal about it until finally, in 1704, he published ''Opticks''. In the meantime the German mathematician [[Gottfried Wilhelm Leibniz]] had developed his own very similar version of the calculus. Both mathematicians used similar ideas of [[infinitesimal]]s to smooth out details of division by zero and other seeming mathematical obstacles.  

Although Newton had communicated his discoveries in the calculus privately, he did not publish anything formal about it until finally, in 1704, he published ''Opticks''. In the meantime the German mathematician [[Gottfried Wilhelm Leibniz]] had developed his own very similar version of the calculus. Although he acknowledged that Newton was earlier, a nasty priority conflict broke out in the 1710s. Newton and his (mainly English) followers accused Leibniz of plagiarism, and the Germans retaliated in kind. The modern view is that both mathematicians discovered the calculus independently. The symbolism in modern use comes from Leibnitz and 18th century French mathematicians.

Although Leibniz acknowledged that Newton was earlier, a nasty priority conflict broke out in the 1710s. Newton and his (mainly English) followers accused Leibniz of plagiarism, and the Germans retaliated in kind. The modern view is that both mathematicians discovered the calculus independently. The symbolism in modern use comes from Leibniz and 18th century French mathematicians.

===Optics===

===Optics===