He was so prolific and original that some of his workwent unnoticed (for example, Weierstrass became famous for showinga nowhere-differentiable continuous function;later it was found that Riemann had casually mentioned onein a lecture years earlier).
When he followed up Augustus de Morgan's earlier work in symboliclogic, de Morgan insisted that Boole was the true master of that field,and begged his friend to finally study mathematics at university.
Dozens of discoveries are named after Poisson; for examplethe Poisson summation formula which has applications in analysis,number theory, lattice theory, etc.
In addition to important advances in several areas of physics,Poisson made key contributions to Fourier analysis, definiteintegrals, path integrals, statistics, partial differential equations,calculus of variations and other fields of mathematics.
Jakob Steiner made many major advances in synthetic geometry, hoping thatclassical methods could avoid any need for analysis;and indeed, like Isaac Newton, he was often able to equal or surpass methodsof analysis or the calculus of variations using just pure geometry;for example he had pure synthetic proofs for a notable extension to Pascal's Mystic Hexagram,and a reproof of Salmon's Theorem that cubic surfaces have exactly 27 lines.
(For example, he showed that all integers except 4k(8m+7)can be expressed as the sum of three squares.)He also made key contributions in several areas of analysis:he invented the Legendre transform and Legendre polynomials;the notation for partial derivatives is due to him.
He also invented the concept of generating functions; for example,letting p(n) denote the number of partitions of n, Euler found the lovely equation: Σn (n) xn= 1 / Πk (1 - xk)
The denominator of the right side hereexpands to a series whose exponents all have the (3m2+m)/2"pentagonal number" form; Euler found an ingenious proof of this.
He was noted for his strong belief in determinism, famously replying toNapoleon's question about God with: "I have no need of that hypothesis."Laplace viewed mathematics as just a tool for developing hisphysical theories.
I won't try to summarize Leibniz' contributions to philosophyand diverse other fields including biology; as justthree examples: he predicted the Earth's molten core,introduced the notion of subconscious mind,and built the first calculator that could do multiplication.
For example, the Law of the Pendulum, based on Galileo's incorrectbelief that the tautochrone was the circle, conflictedwith his own observations.)Despite his extreme importance to mathematical physics,Galileo doesn't usually appear onlists of greatest .
Dirichlet was preeminent in algebraic and analyticnumber theory, but did advanced work in several other fields as well:He discovered the modern definition of function,the Voronoi diagram of geometry, and important concepts indifferential equations, topology, and statistics.
(Some of his designs, including the viola organista, hisparachute, and a large single-span bridge, were finally built fivecenturies later.)He developed the mechanical theory of the arch;made advances in anatomy, botany, and other fields of science;he was first to conceive of plate tectonics.
of all time."His book studiedmultivariate polynomials and is consideredthe best mathematics in ancient China and describes methods notrediscovered for centuries; for exampleZhu anticipated the Sylvester matrix method for solving simultaneouspolynomial equations.
His principal treatise was a letter he wrotethe night before his fatal duel, of whichHermann Weyl wrote: "This letter, if judged by the novelty and profundityof ideas it contains, is perhaps the most substantial piece of writing inthe whole literature of mankind."Galois' ideas were very far-reaching; for example he iscredited as first to prove that trisecting a general angle withPlato's rules is impossible.