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5. Gottfried Wilhelm Riemann
Bernhard Riemann was a remarkable genius whose work was both highly original and rigorous. He had poor physical health and died at an early age, but still made revolutionary contributions in many areas of mathematics. He applied topology to analysis, and applied analysis to number theory. His single paper on the Prime Number distribution conjecture is considered the most important ever on that frequently studied topic. He introduced the clarifying notion of the Riemann integral. He posed the ``Hypothesis of Riemann's zeta function,'' which is regarded as the most important and famous unsolved problem in mathematics still to this day. His masterpieces were differential geometry, tensor analysis, non-Euclidean geometry, the theory of functions, and, especially, the theory of manifolds. He generalized the notions of distance and curvature and, therefore, described new possibilities for the geometry of space itself. Like the greatest mathematicians (Gauss, Archimedes, Newton), Riemann was intensely interested in physics. Although his theory unifying electricity, magnetism and light was supplanted by Maxwell's theory, Riemann Geometry played a significant role in Einstein's Theory of Relativity. And thus, plays a major part in modern physics. It played a major role in this theory because Riemann Geometry states that the triangle's angles > 180̊. This allows lines to cross the same line at 90̊ and cross each other at a circle's pole. For example all longitude lines cross the equator at 90̊ and cross at the north and south poles. If outer space were a black hole or huge circle, space would be curved. This would support Einstein's Theory of Relativity. 4. Leonhard Euler
 Leonard Euler is perhaps the most prolific mathematician to have ever lived. He has made decisive contributions in virtually every area of mathematics. He gave the world modern trigonometry, and just as Archimedes extended Euclid's geometry to marvelous heights, so Euler took marvelous advantage of the analysis of Newton and Leibniz; he made important advances in mathematical physics. Two of the most important advances in 18th century were Lagrange's calculus of variations and Fourier's spectral series and in each case the key initial discovery was actually Euler's. His colleagues called him ``Analysis Incarnate.'' He was supreme at discrete mathematics, as well as continuous: He invented graph theory and generating functions, any one of which would qualify him to be considered for this list. An indication of his importance is that four of the most important constant symbols in mathematics (pi, e, i = sqrt(-1), and gamma = 0.57721566...) were all introduced or popularized by Euler.
Euler combined his brilliance with phenomenal concentration. He developed the first method to estimate the Moon's orbit (the ``three-body problem'' which had stumped Newton), and he settled an arithmetic dispute involving 50 decimal places of a long convergent series. Both these feats were accomplished when he was totally blind. As a young man, Euler discovered and proved the following: pi2/6 = 1 + 1/4 + 1/9 + 1/16 + 1/25 + ... This striking identity catapulted Euler to instant fame, since the right-side infinite sum was a famous unsolved problem of the day. Another equation for which Euler is famous is e^i x = cos x + i sin x. This equation would lead us to Euler’s famour idenity: e^i π +1 = 0. This identity is considered by many to be remarkable for its pure mathematical beauty. Three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also amazingly links five fundamental mathematical constants: the number 0, the number 1, pi, the number e ( which is the base of natural logarithms, which occurs widely in mathematical analysis), and the imaginary number i. Philosophers of mathematics are still not sure what this identity really means, but as Stanford mathematics professor Keith Devlin once said, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence.
3. Sir Isaac Newton
As far as mathematical achievements are concerned however, Newton doesn’t come up short. Although Leibniz also developed the techniques independently, Newton is regarded as the Father of Calculus, his most crucial insight being what is now called the Fundamental Theorem of Calculus (that integration and differentiation are each other's inverse operation). He applied calculus to solve a variety of problems: finding areas, tangents, the lengths of curves and the maxima and minima of functions. Other mathematical works include the Binomial Theorem and the numeric Method which still bears his name. In 1687 Newton published Philosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The motion of the planets was not understood before Newton, and in Principia Newton analyzed the consequences of his Laws of Motion and introduced the Law of Universal Gravitation. The notion that the Earth rotated about the Sun was first introduced by Eudoxus of Cnidus, but Newton explained why it did, and the Great Scientific Revolution had begun. Undoubtedly Newton is one of the most influential people to have ever lived, and would certainly rank at or near the top on any list of physicists, or scientists in general, but I've slightly demoted him on this list as his emphasis was on physics and not mathematics, and Leibniz's contribution lessens the historical importance of Newton's calculus. But even still, as Leibniz once wrote ``Taking mathematics from the beginning of the world to the time when Newton lived, what he has done is much the better part.'' 2. Archimedes of Syracuse
 It is telling that the most prestigious prize in mathematics, the Fields Medal, carries a portrait of Archimedes of Syracuse, along with his proof concerning the sphere and the cylinder. Undoubted one of the most brilliant minds to have ever walked the face of the earth, Archimedes was a Greek mathematician, physicist, engineer, inventor, and astronomer. It is thought that he ay have studied at Euclid's school, but his work far surpassed the works of Euclid. Archimedes made major advances in number theory, algebra, and geometry and his methods anticipated both the integral and differential calculus. His achievements are particularly impressive given the lack of good mathematical notation in his day.
His original achievements in physics include the principles of leverage, the first law of hydrostatics, and inventions like the compound pulley, the hydraulic screw, and numerous war machines. His works include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder. Archimedes had proved that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. His tomb carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. Recently, modern technology has led to the discovery of new writings by Archimedes, hitherto hidden on a palimpsest. This has caused Archimedes to rise even higher in the esteem of mathematical historians. These new writings imply an understanding of the distinction between countable and uncountable infinities, a distinction which wasn't resolved until Georg Cantor, who lived 2300 years after the time of Archimedes. 1. Carl Fredrich Gauss
 Although every single mathematician on our list is a first rate genius, there is one who is so far above all the others that there can be no argument whatsoever that he is the greatest mathematician of all time. Carl Friedrich Gauss, the ``Prince of Mathematics,'' first exhibited his calculative powers when he corrected his father's arithmetic at the age of three. At the age of eight, a teacher gave his students an assignment to add up the first 100 numbers. Instantly, Gauss said that he had completed the exercise (the story goes that he had figured that 100 numbers could be determined by the equation n(a+b)(1/2)=50(a+b) where n=100, a = the first digit in the sequence and b = the last digit in the sequence.) At twelve, his revolutionary nature was demonstrated when he began questioning the axioms of Euclid. And his genius was confirmed at the age of nineteen when he proved that the regular n-gon was constructible if and only if n is the product of prime Fermat numbers. At age 24 he published Disquisitiones Arithmeticae, which is unquestionably the greatest book of pure mathematics eve written.
Gauss’s contributions to mathematical knowledge are so vast that any attempt to summarize them ina few short paragraphs would be a futile gesture. He built the theory of complex numbers into its modern form, including the notion of ``monogenic'' functions which are now ubiquitous in mathematical physics. Gauss’s other contributions are quite numerous, including proving the Fundamental Theorem of Arithmetic, the Fundamental Theorem of Algebra (an n-th degree polynomial has n complex roots), the foundations of statistics (including Law of Least Squares), differential geometry. He is universally considered to be the premier number theoretician of all time, having proved Euler's Law of Quadratic Reciprocity. He also did important work in several areas of physics. Much of Gauss's work wasn't published: unbeknownst to his colleagues it was Gauss who first discovered doubly periodic elliptic functions, non-Euclidean geometry, quaternions, the foundations of topology, the ``butterfly'' procedure for rapid calculation of Fourier series, and even the rudiments of knot theory. Also in this category is the Fundamental Theorem of Functions of a Complex Variable (that the line-integral over a closed curve of a monogenic function is zero): he proved this first but let Cauchy take the credit. a
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