Blaise Pascal, 1623-1662, was one of the most famous child protégés that ever lived. (At 12 he discovered many geometry theorems on his own, being forbidden to study the subject. At 16 he wrote a book on conic sections.) He not only discovered Pascal's Principal in Physics, but the cycloid, and was known for his writings in Philosophy such as "The Wager," which commented on the afterlife and existence of God.
Kummer was the one who introduced unique factorization into algebraic numbers and advanced Fermat's last Theorem.
Gauss: The Prince of Mathematicians; he first computed the movements of the asteroid Ceres, and discovered the Bell curve, as is shown on a 10 Mark German note.
Archimedes completed an infinite series, found the value of pi to two decimals, discovered a way to integrate the parabola using an infinite series of triangles**. Oh yes! He also trisected the angle by way of a paper strip.
Fermat, 1601-1665, a complete amateur in mathematics and a jurist by trade, was unbelievably ahead of his time in Number Theory. He not only knew that every integer is the sum of four squares, but also that evey integer is the sum of three triangle numbers, of five pentangle, etc. He completed a whopper in one fell swoop*. He and Pascal introduced the theory of probability through a series of personal correspondence. He is know in Physics for the fact that light from one medium to another travels the path of least time.
*Note: A theorem also known as Bachet's conjecture which was stated but not proven by Diophantus. It states that every positive integer can be written as the sum of at most four squares. Although the theorem was proved by Fermat using infinite descent, the proof was suppressed. Euler was unable to prove the theorem. The first published proof was given by Lagrange in 1770 (more than 100 years after Fermat died) and made use of the Euler four-square identity.
http://mathworld.wolfram.com/LagrangesFour-SquareTheorem.html
**In Archimedes day, of course, there was no Cartesian coordinate system. His method of computing the area of a parabola was to find a series of triangles,doubling in number and decreasing in size, with a fixed ratio of A= area of largest then: A/8, A/64...etc. This was suitable to the use of plane geometry. The modern method using rectangles and of general use for continuous functions was, of course, unknown to him. So that it is probably not proper to speak of him as a discoverer of the Calculus, as a few have suggested; but he did have an understand of the limit process of converging infinite sums.
