Newton was an industrious lad who built marvelous toys (e.g. a model windmill powered by a mouse on treadmill). At about age 22, on leave from University, this genius began revolutionary advances in mathematics, optics, dynamics, thermodynamics, acoustics and celestial mechanics. He is most famous for his Three Laws of Motion (inertia, force, reciprocal action) and Law of Universal Gravitation. As Newton himself acknowledged, the Laws weren't fully novel: Hipparchus, Ibn al-Haytham, Galileo and Huygens had all developed much basic mechanics already, and Newton credits the First Law itself to Aristotle. (However, since Christiaan Huygens, the other great mechanist of the era and who had also deduced that Kepler's laws imply inverse-square gravitation, considered the action at a distance in Newton's universal gravitation to be "absurd," at least this much of Newton's mechanics must be considered revolutionary. Newton's other intellectual interests included chemistry, theology, astrology and alchemy.) Although this list is concerned only with mathematics, Newton's greatness is indicated by the wide range of his physics: even without his revolutionary Laws of Motion and his Cooling Law of thermodynamics, he'd be famous just for his work in optics, where he explained diffraction and observed that white light is a mixture of all the rainbow's colors. (Although his corpuscular theory competed with Huygen's wave theory, Newton understood that his theory was incomplete without waves, and thus anticipated wave-particle duality.) Newton also designed the first reflecting telescope, first reflecting microscope, and the sextant.

Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (which he called "fluxions"); he shares credit with Leibniz for the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus for several purposes: finding areas, tangents, the lengths of curves and the maxima and minima of functions. In addition to several other important advances in analytic geometry, his mathematical works include the Binomial Theorem, his eponymous numeric method, the idea of polar coordinates, and power series for exponential and trigonometric functions. (His equationhas been called the "most important series in mathematics.") He contributed to algebra and the theory of equations, generalizing Déscartes' rule of signs. (The generalized rule of signs was incomplete and finally resolved two centuries later by Sturm and Sylvester.) He developed a series for the arcsin function. He developed facts about cubic equations (just as the "shadows of a cone" yield all quadratic curves, Newton found a curve whose "shadows" yield all cubic curves). He proved that same-mass spheres of any radius have equal gravitational attraction: this fact is key to celestial motions. He discovered Puiseux series almost two centuries before they were re-invented by Puiseux. (Like some of the greatest ancient mathematicians, Newton took the time to compute an approximation toe^{x}=∑x^{k}/ k!π; his was better than Vieta's, though still not as accurate as al-Kashi's.)

Newton is so famous for his calculus, optics and laws of motion, it is easy to overlook that he was also one of the greatest geometers. He solved the Delian cube-doubling problem. Even before the invention of the calculus of variations, Newton was doing difficult work in that field, e.g. his calculation of the "optimal bullet shape." Among many marvelous theorems, he proved several about quadrilaterals and their in- or circum-scribing ellipses, and constructed the parabola defined by four given points. He anticipated Poncelet's Principle of Continuity. An anecdote often cited to demonstrate his brilliance is the problem of thebrachistochrone, which had baffled the best mathematicians in Europe, and came to Newton's attention late in life. He solved it in a few hours and published the answer anonymously. But on seeing the solution Jacob Bernoulli immediately exclaimed "I recognize the lion by his footprint."

In 1687 Newton publishedPhilosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The motion of the planets was not understood before Newton, although theheliocentricsystem allowed Kepler to describe the orbits. InPrincipiaNewton analyzed the consequences of his Laws of Motion and introduced the Law of Universal Gravitation. (In this work Newton also proved important theorems about inverse-cubeforces, work largely unappreciated until Chandrasekhar's modern-day work.) The notion that the Earth rotated about the Sun was introduced in ancient Greece, but Newton explainedwhyit did, and the Great Scientific Revolution began. Newton once wrote "Truth is ever to be found in the simplicity, and not in the multiplicity and confusion of things." Sir Isaac Newton was buried at Westminster Abbey in a tomb inscribed "Let mortals rejoice that so great an ornament to the human race has existed."

Newton ranks #2 on Michael Hart's famous list of the Most Influential Persons in History. (Muhammed the Prophet of Allah is #1.) Whatever the criteria, Newton would certainly rank first, or behind only Einstein, on any list of physicists, or scientists in general, but some listmakers would demote him slightly on a list of pure mathematicians: his emphasis was physics not mathematics, and the contribution of Leibniz (Newton's rival for the titleInventor of Calculus) lessens the historical importance of Newton's calculus. One reason I've ranked him at #1 is a comment by Gottfried Leibniz himself: "Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part."

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