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The Fifteen Greatest Mathematicians

  1. Carl F. Gauss

  2. Leonhard Euler

  3. Archimedes  of Syracuse

  4. Sir Isaac Newton

  5. Euclid of Alexandria

  6. Bernhard Riemann

  7. Gottfried Wilhelm Leibniz

  8. Carl G. J. Jacobi

  9. Henri Poincaré

10. Pierre de Fermat

11. Augustin Cauchy

12. Niels Abel

12. Srinivasa Ramanujan  Iyengar

14. Évariste Galois

15. Blaise Pascal

 

Here are the Fifteen Greatest Mathematicians in chronological order:
 

Euclid of Alexandria (ca. 322 – ca. 275 BC) Greece/Egypt

Euclid may have been a student of Aristotle. He founded the school of mathematics at the great university of Alexandria. He was the first to prove that there are infinitely many prime numbers, and established the relationship between perfect numbers and Mersenne primes. Among several books attributed to him are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), and his comprehensive math textbook The Elements. Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. The Elements introduced the notions of axiom and theorem. It was used as a textbook for 2000 years and in fact is still the basis for high school geometry, making Euclid the leading mathematics teacher of all time. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem.

Archimedes of Syracuse (287-212 BC) Greece

Archimedes studied at Euclid's school (probably after Euclid's death), but his work far surpassed the works of Euclid. Archimedes made advances in number theory and algebra, but his greatest contributions were in geometry. His methods anticipated both the integral and differential calculus. 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 achievements are particularly impressive given the lack of good mathematical notation in his day.

His works include Floating Bodies, Spirals, The Sand Reckoner, Measurement of the Circle, and Sphere and Cylinder. 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.

Blaise Pascal (1623-1662) France

Pascal was an outstanding genius who studied geometry as a child. At the age of sixteen he stated and proved Pascal's Theorem, a fact relating any six points on any conic section. The Theorem is sometimes called the “Cat's Cradle'' or the “Mystic Hexagram.'' While most of the important theorems of mathematics would have been discovered anyway, just a few years delayed, had their genius creator never existed, one wonders how long Pascal's marvelous result would have gone undiscovered, without this amazing teenage prodigy. In addition to classic and projective geometry, Pascal founded probability theory, made contributions to axiomatic theory, and the invention of calculus. He may be most famous for his description of Pascal's Triangle.

Like most of the greatest mathematicians, he was interested in physics and mechanics, studying fluids, explaining vacuum, and inventing the syringe and hydraulic press. At the age of eighteen Pascal designed and built the world's first automatic adding machine. (Although he continued to refine this invention, it was never a commercial success.)

Pascal abandoned mathematics for religion, suffered poor health, and died at an early age.

Isaac  (Sir)  Newton (1642-1727) England

Newton was an industrious lad who built marvelous toys. His genius seems to have blossomed at about age 22 when, on leave from University, he began revolutionary advances in mathematics, optics, dynamics, and celestial mechanics. Newton's other intellectual interests included theology and alchemy.

Although others also developed the techniques independently, Newton is regarded as the Father of Calculus (what he called the “method of fluxions''); 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. An anecodote 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 Johann 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 analysed 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 the Eudoxus of Cnidus mentioned above, but Newton explained why it did, and the Great Scientific Revolution began.

Newton would certainly rank at the top on any list of physicists, or scientists in general, but I've demoted him on this list: his emphasis was physics not mathematics, and Leibniz's contribution lessens the historical importance of Newton's calculus. A comment by Leibniz, however, persuades me not to rank Newton any worse than 4th place: Despite being a rival for the title of <I<>Inventor of Calculus, Leibniz stated that this great genius “… advanced mathematics probably more than all before him combined.''

Srinivasa Ramanujan Iyengar (1887-1920) India

Like Abel, Ramanujan was a self-taught prodigy who lived in a country distant from his mathematical peers, and suffered from poverty: childhood dysentery and vitamin deficiencies probably led to his early death. Yet he produced 4000 theorems or conjectures in number theory, algebra, and combinatorics. His specialties included infinite series, elliptic functions, continued fractions, partition enumeration, definite integrals, modular equations, and “highly composite'' numbers. His innate ability for algebraic manipulations equalled or surpassed that of Euler and Jacobi. Although many formulae have been discovered to calculate pi, a bizarre formula of Ramanujan is often used, because of its fast convergence. Many of Ramanujan's results would probably never have been discovered without him, and are so inspirational that there is a periodical dedicated to them. The theories of strings and crystals have benefited from Ramanujan's work. (Today some professors “make their name'' just by finding a new proof for one of Ramanujan's many results.)

Unlike Abel, who insisted on rigorous proofs, Ramanujan often omitted proofs. Unlike Abel, most of whose work specifically depended on the complex numbers, Ramanujan mostly worked only with real numbers. Despite these limitations, Ramanujan is considered one of the greatest geniuses ever.

 

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