Leonhard Euler (1707-1783) Switzerland
Just as Archimedes extended Euclid's geometry to marvelous heights, so Euler took marvelous advantage of the analysis of Newton and Leibniz.
He gave the world modern trigonometry. Along with Lagrange he pioneered the calculus of variations. He was also supreme at discrete mathematics, inventing graph theory and generating functions. Euler was also a major figure in number theory, proving that the sum of the reciprocals of primes less than x is approx. (ln ln x), finding both the largest then-known prime and the largest then-known perfect number, proving e to be irrational, proving that all even perfect numbers must have the Mersenne number form that Euclid had discovered 2000 years earlier, and much more. Euler was also first to prove several interesting theorems of geometry, including facts about the 9-point Feuerbach circle; relationships among a triangle's altitudes, medians, and circumscribing and inscribing circles; and an expression for a tetrahedron's area in terms of its sides. Euler was first to explore topology, proving theorems about the Euler characteristic. He also made several important advances in physics, e.g. extending Newton's Laws of Motion to rotating rigid bodies. On a lighter note, Euler constructed a particularly "magical" magic square.
Four of the most important constant symbols in mathematics (π, e, i = √-1, and γ = 0.57721566...) were all introduced or popularized by Euler. He did important work with Riemann's zeta function ζ(s) = ∑ k-s (although it was not then known with that name or notation); he anticipated the concept of analytic continuation by "proving" ζ(-1) = 1+2+3+4+... = -1/12. As a young student of the Bernoulli family, Euler discovered the striking identity π2/6 = ζ(2) This catapulted Euler to instant fame, since the right-side infinite sum (1 + 1/4 + 1/9 + 1/16 + ...) was a famous problem of the time. Among many other famous and important identities, Euler proved the Pentagonal Number Theorem (a beautiful little result which has inspired a variety of discoveries), and the Euler Product Formula ζ(s) = ∏(1-p-s)-1 where the right-side product is taken over all primes p. His most famous identity (which Richard Feynman called an "almost astounding ... jewel") unifies the trigonometric and exponential functions: ei x = cos x + i sin x.
Some of Euler's greatest formulae can be combined into curious-looking formulae for π: π2 = - log2(-1) = 6 ∏p∈Prime(1-p-2)-1/2
Gottfried Wilhelm von Leibniz (1646-1716) Germany
Mathematics was just a self-taught sideline for Leibniz, who was a philosopher, lawyer, historian, diplomat and renowned inventor. Because he "wasted his youth" before learning mathematics, he probably ranked behind the Bernoullis as well as Newton in pure mathematical talent, and thus he may be the only mathematician among the Top Ten who was never the greatest living algorist or theorem prover. We won't try to summarize Leibniz' contributions to philosophy and diverse other fields including biology; as just three examples: he predicted the Earth's molten core, introduced the notion of subconscious mind, and built the first calculator that could do multiplication.
Leibniz pioneered the common discourse of mathematics, including its continuous, discrete, and symbolic aspects. (His ideas on symbolic logic weren't pursued and it was left to Boole to reinvent this almost two centuries later.) Mathematical innovations attributed to Leibniz include the symbols ∫, df(x)/dx; the concepts of matrix determinant and Gaussian elimination; the theory of geometric envelopes; and the binary number system. He invented more mathematical terms than anyone, including "function," "analysis situ," "variable," "abscissa," "parameter," and "coordinate." His works seem to anticipate cybernetics and information theory; and Mandelbrot acknowledged Leibniz' anticipation of self-similarity. Like Newton, Leibniz discovered The Fundamental Theorem of Calculus; his contribution to calculus was much more influential than Newton's, and his superior notation is used to this day. As Leibniz himself pointed out, since the concept of mathematical analysis was already known to ancient Greeks, the revolutionary invention was notation ("calculus"), because with "symbols [which] express the exact nature of a thing briefly ... the labor of thought is wonderfully diminished."
Although others found it independently (including perhaps Madhava three centuries earlier), Leibniz discovered and proved a striking identity for π:
π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...
Joseph-Louis (Comte de) Lagrange (1736-1813) Italy, France
He proved a fundamental Theorem of Group Theory. He laid the foundations for the theory of polynomial equations which Cauchy, Abel, Galois and Poincaré would later complete. Number theory was almost just a diversion for Lagrange, whose focus was analysis; nevertheless he was the master of that field as well, proving difficult and historic theorems including Wilson's theorem (p divides (p-1)! + 1 when p is prime); Lagrange's Four-Square Theorem (every positive integer is the sum of four squares); and that n•x2 + 1 = y2 has solutions for every positive non-square integer n.
Lagrange's many contributions to physics include understanding of vibrations (he found an error in Newton's work and published the definitive treatise on sound), celestial mechanics (including an explanation of why the Moon keeps the same face pointed towards the Earth), the Principle of Least Action (which Hamilton compared to poetry), and the discovery of the Lagrangian points (e.g., in Jupiter's orbit). Unlike Newton, who used calculus to derive his results but then worked backwards to create geometric proofs for publication, Lagrange relied only on analysis. "No diagrams will be found in this work" he wrote in the preface to his masterpiece Mécanique analytique.
Lagrange once wrote "As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection." Both W.W.R. Ball and E.T. Bell, renowned mathematical historians, bypass Euler to name Lagrange as "the Greatest Mathematician of the 18th Century." Jacobi bypassed Newton and Gauss to call Lagrange "perhaps the greatest mathematical genius since Archimedes."
Jean Baptiste Joseph Fourier (1768-1830) France
Joseph Fourier had a varied career: precocious but mischievous orphan, theology student, young professor of mathematics (advancing the theory of equations), then revolutionary activist. Under Napoleon he was a brilliant and important teacher and historian; accompanied the French Emperor to Egypt; and did excellent service as district governor of Grenoble. In his spare time at Grenoble he continued the work in mathematics and physics that led to his immortality.
After the fall of Napoleon, Fourier exiled himself to England, but returned to France when offered an important academic position and published his revolutionary treatise on the Theory of Heat. Fourier anticipated linear programming, developing the simplex method and Fourier-Motzkin Elimination. He is also noted for the notion of dimensional analysis, was first to describe the Greenhouse Effect, and continued his earlier brilliant work with equations.
Fourier's greatest fame rests on his use of trigonometric series (now called Fourier series) in the solution of differential equations. Since "Fourier" analysis is in extremely common use among applied mathematicians, he joins the select company of the eponyms of "Cartesian" coordinates, "Gaussian" curve, and "Boolean" algebra. Because of the importance of Fourier analysis, many listmakers would rank Fourier much higher than I have done; however the work was not exceptional as pure mathematics. Fourier's Heat Equation built on Newton's Law of Cooling; and the Fourier series solution itself had already been introduced by Euler, Lagrange and Daniel Bernoulli. Fourier's solution to the heat equation was counterintuitive (heat transfer doesn't seem to involve the oscillations fundamental to trigonometric functions): The brilliance of Fourier's imagination is indicated in that the solution had been rejected by Lagrange himself. Although rigorous Fourier Theorems were finally proved only by Dirichlet, Riemann and Lebesgue, it has been said that it was Fourier's "very disregard for rigor" that led to his great achievement, which Lord Kelvin compared to poetry.
Niels Henrik Abel (1802-1829) Norway
At an early age, Niels Abel studied the works of the greatest mathematicians, found flaws in their proofs, and resolved to reprove some of these theorems rigorously. He was the first to fully prove the general case of Newton's Binomial Theorem, one of the most widely applied theorems in mathematics.
Perhaps his most famous achievement was the (deceptively simple) Abel's Theorem of Convergence (published posthumously), one of the most important theorems in analysis; but there are several other Theorems which bear his name. Abel also made contributions in algebraic geometry and the theory of equations.
Inversion (replacing y = f(x) with x = f-1(y)) is a key idea in mathematics (consider Newton's Fundamental Theorem of Calculus); Abel developed this insight. Legendre had spent much of his life studying elliptic integrals, but Abel inverted these to get elliptic functions, which quickly became a productive field of mathematics, and led to more general complex-variable functions, which were important to the development of both abstract and applied mathematics.
Finding the roots of polynomials is a key mathematical problem: the general solution of the quadratic equation was known by ancients; the discovery of general methods for solving polynomials of degree three and four is usually treated as the major math achievement of the 16th century; so for over two centuries an algebraic solution for the general 5th-degree polynomial (quintic) was a Holy Grail sought by most of the greatest mathematicians. Abel proved that most quintics did not have such solutions. This discovery, at the age of only nineteen, would have quickly awed the world, but Abel was impoverished, had few contacts, and spoke no German. When Gauss received Abel's manuscript he discarded it unread, assuming the unfamiliar author was just another crackpot trying to square the circle or some such. His genius was too great for him to be ignored long, but, still impoverished, Abel died of tuberculosis at the age of twenty-six. His fame lives on and even the lower-case word 'abelian' is applied to several concepts. Hermite said "Abel has left mathematicians enough to keep them busy for 500 years."
Srinivasa Ramanujan Iyengar (1887-1920) India
Much of his methodology, including unusual ideas about divergent series, was his own invention. (As a young man he made the absurd claim that 1+2+3+4+... = -1/12. Later it was noticed that this claim translates to a true statement about the Riemann zeta function, with which Ramanujan was unfamiliar.) Ramanujan's innate ability for algebraic manipulations equaled or surpassed that of Euler and Jacobi.
Ramanujan's most famous work was with the partition enumeration function p(), Hardy guessing that some of these discoveries would have been delayed at least a century without Ramanujan. Together, Hardy and Ramanujan developed an analytic approximation to p(). (Rademacher and Selberg later discovered an exact expression to replace the Hardy-Ramanujan formula; when Ramanujan's notebooks were studied it was found he had anticipated their technique, but had deferred to his friend and mentor.)
The theories of strings and crystals have benefited from Ramanujan's work. (Today some professors achieve fame just by finding a new proof for one of Ramanujan's many results.) Unlike Abel, who insisted on rigorous proofs, Ramanujan often omitted proofs. (Ramanujan may have had unrecorded proofs, poverty leading him to use chalk and erasable slate rather than paper.) Unlike Abel, much of whose work depended on the complex numbers, most of Ramanujan's work focused on real numbers. Despite these limitations, Ramanujan is considered one of the greatest geniuses ever.
Because of its fast convergence, an odd-looking formula of Ramanujan is often used to calculate π:
992 / π = √8 ∑k=0,∞ (4k! (1103+26390 k) / (k!4 3964k))
By Navina!
Archimedes
Archimedes’s methods anticipated both the integral and differential calculus.
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. Archimedes proved that the volume of a sphere is two-thirds the volume of a 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.
Isaac Newton
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.
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. He developed a series for the arcsin function. He developed facts about cubic equations and 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.
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. 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."
Carl F. Gauss
Carl Friedrich Gauss, the "Prince of Mathematics," exhibited his calculative powers when he corrected his father's arithmetic before the age of three. His revolutionary nature was demonstrated at age twelve, when he began questioning the axioms of Euclid. His genius was confirmed at the age of nineteen when he proved that the regular n-gon was constructible, for odd n, if and only if n is the product of distinct prime Fermat numbers. At age 24 he published Disquisitiones Arithmeticae, probably the greatest book of pure mathematics ever.
Several important theorems and lemmas bear his name; he was first to produce a complete proof of Euclid's Fundamental Theorem of Arithmetic (that every natural number has a unique expression as product of primes); and first to produce a rigorous proof of the Fundamental Theorem of Algebra. Gauss himself used "Fundamental Theorem" to refer to Euler's Law of Quadratic Reciprocity; Gauss was first to provide a proof for this, and provided eight distinct proofs for it over the years. Gauss proved the n=3 case of Fermat's Last Theorem for a class of complex integers; though more general, the proof was simpler than the real integer proof, a discovery which revolutionized algebra.
Other work by Gauss led to fundamental theorems in statistics, vector analysis, function theory, and generalizations of the Fundamental Theorem of Calculus.
Gauss built the theory of complex numbers into its modern form, including the notion of "monogenic" functions which are now ubiquitous in mathematical physics. Gauss was the premier number theoretician of all time, Other contributions of Gauss include hypergeometric series, foundations of statistics, and differential geometry. He also did important work in geometry, providing an improved solution to Apollonius' famous problem of tangent circles, stating and proving the Fundamental Theorem of Normal Axonometry, and solving astronomical problems related to comet orbits and navigation by the stars. (The first asteroid was discovered when Gauss was a young man; he famously constructed an 8th-degree polynomial equation to predict its orbit.) Gauss also did important work in several areas of physics, and invented the heliotrope.
Much of Gauss's work wasn't published: unbeknownst to his colleagues it was Gauss who first discovered non-Euclidean geometry, doubly periodic elliptic functions, a prime distribution formula, quaternions, foundations of topology, the Law of Least Squares, Dirichlet's class number formula, the key Bonnet's Theorem of differential geometry (now usually called Gauss-Bonnet Theorem), 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.
Euclid
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; he stated and proved the unique factorization theorem; and he devised Euclid's algorithm for computing gcd. He introduced the Mersenne primes and observed that (M2+M)/2 is always perfect if M is Mersenne. He proved that there are only five "Platonic solids," as well as theorems of geometry far too numerous to summarize; among many with special historical interest is the proof that rigid-compass constructions can be implemented with collapsing-compass constructions. Among several books attributed to Euclid are The Division of the Scale (a mathematical discussion of music), The Optics, The Cartoptrics (a treatise on the theory of mirrors), a book on spherical geometry, a book on logic fallacies, and his comprehensive math textbook The Elements.
Several of his masterpieces have been lost, including works on conic sections and other advanced geometric topics. Apparently Desargues' Homology Theorem (a pair of triangles is coaxial if and only if it is copolar) was proved in one of these lost works; this is the fundamental theorem which initiated the study of projective geometry. Some think his best inspiration was recognizing that the Parallel Postulate must be an axiom rather than a theorem.
There are many famous quotations about Euclid and his books. Abraham Lincoln abandoned his law studies when he didn't know what "demonstrate" meant and "went home to my father's house [to read Euclid], and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies."
David Hilbert
Hilbert was pre-eminent in many fields of mathematics, including axiomatic theory, invariant theory, algebraic number theory, class field theory and functional analysis. His examination of calculus led him to the invention of "Hilbert space," considered one of the key concepts of functional analysis and modern mathematical physics. He was a founder of fields like metamathematics and modern logic. He was also the founder of the "Formalist" school which opposed the "Intuitionism" of Kronecker and Brouwer. He developed a new system of definitions and axioms for geometry, replacing the 2200 year-old system of Euclid. As a young Professor he proved his "Finiteness Theorem," now regarded as one of the most important results of general algebra. The methods he used were so novel that, at first, the "Finiteness Theorem" was rejected for publication as being "theology" rather than mathematics! In number theory, he proved Waring's famous conjecture which is now known as the Hilbert-Waring theorem.
Any one man can only do so much, so the greatest mathematicians should help nurture their colleagues. Hilbert provided a famous List of 23 Unsolved Problems, which inspired and directed the development of 20th-century mathematics. Hilbert was warmly regarded by his colleagues and students, and contributed to the careers of several great mathematicians and physicists including Georg Cantor, Hermann Minkowski, Hermann Weyl, John von Neumann, Emmy Noether, Alonzo Church, and Albert Einstein.
Eventually Hilbert turned to physics and made key contributions to classical and quantum physics and to general relativity. He may have published the "Einstein Field Equations" independently of Einstein. (Since he had already learned of the theory's intuition from personal lectures by Einstein, it is wrong, as some do, to claim Hilbert's publication diminishes Einstein's greatness.)
