A mathematician is a person with an extensive knowledge of mathematics who uses this knowledge in their work, typically to solve mathematical problems. Mathematics is concerned with numbers, data, collection, quantity, structure, space, and change.
Mathematicians involved with solving problems outside of pure mathematics are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of the imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, and localized constructs, applied mathematicians work regularly in the study and formulation of mathematical models. Mathematicians and applied mathematicians are considered to be two of the STEM (science, technology, engineering, and mathematics) careers.
The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on problems, often concrete but sometimes abstract. As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science, engineering, business, and other areas of mathematical .
Archimedes is universally acknowledged to be the greatest of ancient mathematicians. He studied at Euclid's school (probably after Euclid's death), but his work far surpassed the works of Euclid. His achievements are particularly impressive given the lack of good mathematical notation in his day. His proofs are noted not only for brilliance but for unequalled clarity, with a modern biographer (Heath) describing Archimedes' treatises as "without exception monuments of mathematical exposition ... so impressive in their perfection as to create a feeling akin to awe in the mind of the reader." Archimedes made advances in number theory, algebra, and analysis, but is most renowned for his many theorems of plane and solid geometry. He was first to prove Heron's formula for the area of a triangle. His excellent approximation to √3 indicates that he'd partially anticipated the method of continued fractions. He found a method to trisect an arbitrary angle (using a markable straightedge — the construction is impossible using strictly Platonic rules). One of his most remarkable and famous geometric results was determining the area of a parabolic section, for which he offered two independent proofs, one using his Principle of the Lever, the other using a geometric series. Many of Archimedes' discoveries are known only second-hand: Pappus reports that he discovered the Archimedean solids; Thabit ibn Qurra reports his method to construct a regular heptagon; Alberuni credits the Broken-Chord Theorem to him; etc. Archimedes anticipated integral calculus, most notably by determining the centers of mass of hemisphere and cylindrical wedge, and the volume of two cylinders' intersection. Although Archimedes made little use of differential calculus, Chasles credits him (along with Kepler, Cavalieri, and Fermat) as one of the four who developed calculus before Newton and Leibniz. He was similar to Newton in that he used his (non-rigorous) calculus to discover results, but then devised rigorous geometric proofs for publication. 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 war machines. His books include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder. He developed the Stomachion puzzle (and solved a difficult enumeration problem involving it). Archimedes proved that the volume of a sphere is two-thirds the volume of a circumscribing cylinder. He requested that a representation of such a sphere and cylinder be inscribed on his tomb. Archimedes discovered formulae for the volume and surface area of a sphere, and may even have been first to notice and prove the simple relationship between a circle's circumference and area. For these reasons, π is often called Archimedes' constant. His approximation 223/71 < π < 22/7 was the best of his day. (Apollonius soon surpassed it, but by using Archimedes' method.) That Archimedes shared the attitude of later mathematicians like Hardy and Brouwer is suggested by Plutarch's comment that Archimedes regarded applied mathematics "as ignoble and sordid ... and did not deign to [write about his mechanical inventions; instead] he placed his whole ambition in those speculations the beauty and subtlety of which are untainted by any admixture of the common needs of life." Some of Archimedes' greatest writings are preserved on a palimpsest which has been rediscovered and properly studied only since 1998. Ideas unique to that work are calculating the volume of a cylindrical wedge (previously first attributed to Kepler), and perhaps an implication that Archimedes understood 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). Although Newton may have been the most important mathematician, and Gauss the greatest theorem prover, it is widely accepted that Archimedes was the greatest genius who ever lived. Yet, Hart omits him altogether from his list of Most Influential Persons: Archimedes was simply too far ahead of his time to have great historical significance.
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 famous for his Three Laws of Motion (inertia, force, reciprocal action) but, as Newton himself acknowledged, these 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 Newton was also apparently the first person to conclude that the ordinary gravity we observe on Earth is the very same force that keeps the planets in orbit. His Law of Universal Gravitation was revolutionary and due to Newton alone. (Christiaan Huygens, the other great mechanist of the era, had independently deduced that Kepler's laws imply inverse-square gravitation, but he considered the action at a distance in Newton's theory to be "absurd.") 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.) 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 equation ex = ∑ xk / k! has been called the "most important series in mathematics.") He contributed to algebra and the theory of equations; he was first to state Bézout's Theorem; he generalized 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 to 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 the brachistochrone, 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 published Philosophiae Naturalis Principia Mathematica, surely the greatest scientific book ever written. The motion of the planets was not understood before Newton, although the heliocentric system allowed Kepler to describe the orbits. In Principia Newton analyzed the consequences of his Laws of Motion and introduced the Law of Universal Gravitation. With the key mystery of celestial motions finally resolved, the Great Scientific Revolution began. (In his work Newton also proved important theorems about inverse-cube forces, work largely unappreciated until Chandrasekhar's modern-day work.) 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 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 title Inventor 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."