Banach-Tarski Paradox: In 1924, two Polish mathematicians, Stefan Banach and Alfred Tarski, proved a rather peculiar result: you could decompose a sphere (or any polyhedral figure) into a finite number of pieces, then from those pieces, reconstruct a similar sphere of larger volume. Since this would imply that all volumes are the same, this upset a number of people.
Cantor, Georg: German mathematician who first examined the notion of infinities and showed that not all infinities are equal: some are more equal then others.
Cardano, Giralamo: Italian mathematician, best known for his Ars Magna, a compendium of algebra published in 1545 (right on the heels of Vesalius De Fabrica Corporis Humanis and Copernicus' De Revolutionibus: the 1540s were a banner decade for scientific advance.) Cardano is best known for having published in Ars Magna Niccolo Tartaglia's rule for solving cubic equations, much to Tartaglia's annoyance. (Cardano gave Tartaglia full credit, though not very loudly...in those days, practitioners of mathematics got their fame by being able to solve problems no one else could, and if every Tomas, Riccardo, and Enrico could solve a cubic, Niccolo's reputation would be worthless). Tartaglia spent the rest of his life trying to discredit Cardano.
Cubic Equations: The Greeks had a geometric method of solving general quadratic equations (which can, if you are very, very, very generous, be called the quadratic formula). For thousands of years, though, it was unknown whether a similar method could be found for the general cubic. Omar Khayyam, the Persian geometer and occasional poet, thought it was impossible, for the best reason: he couldn't figure out how to do it. (An Italian named Luca Pacioli thought the same thing, around 1500). Around 1501, an Italian geometer (calling someone who did geometry a mathematician was like calling an astronomer an astrologer, for much the same reason) named Scipione del Ferro discovered a method for solving certain "degenerate" cubics. Somewhat later, Niccolo Tartaglia found the same method. Under oath of secrecy, he told the method of Giralamo Cardano (possibly because he wanted Cardano's support in obtaining a position from Cardano's patron). Using Tartaglia's method, Cardano's student, Lodovico Ferrara, discovered a method of solving the general quartic (fourth degree equation). Finally, Galois and Abel were able to show that solutions to the fifth or higher degree equations were impossible. Solving the cubic was fraught with danger.
Curse of the Cubic: As he lay dying from the sword wounds of a Roman soldier, Archimedes spat a curse to those who would try and solve algebraic problems:
"Lines and planes you may resolve;Actually, he said no such thing. But you might almost think so:
Cubes and others never solve!"
Euler, Leonhard (1707-1783): Swiss mathematician; the most prolific mathematician who ever lived, though if you consider only his works that were pivotal in the development of mathematics, the number reduces itself to a few hundred papers. For a while, Euler was at the court of Catherine the Great (Empress of Russia), along with Rene Diderot and a few other philosophes. Euler was a devout Christian; Diderot was a notorious atheist and encyclopedist. Diderot heard that Euler had a mathematical proof of the existence of God, and asked for it...and got it.
Galois, Evariste: French mathematician who developed the theory of groups; he is most famous for introducing symmetry groups, which lead to the proof that the general fifth degree and higher equations have no general solutions involving finite operations on radicals. For an entertaining history of Galois, read the chapter in Eric Temple Bell's Men of Mathematics. For an accurate history of Galois, read the newer study done by I don't have the reference here.
Gauss, Karl Friedrich: German mathematician, known for his brilliance. At the age of 19, he had to decide on graduate school, and two things interested him: philology (what we would call linguistics today) and mathematics. He made his decision when he determined how to construct, using compass and straightedge alone, a regular septendecagon (a seventeen sided figure).
Infinity: A concept first examined in detail by Georg Cantor. Cantor discovered that while all infinities are equal, some are more equal than others (or was that Orwell who discovered this?). There is an infinity which describes the size of the counting numbers, called aleph-null. There is a larger infinity which describes the size of the real numbers, called C; the proof that these two infinities are not the same is fairly easy. Then there's a third infinity, which describes the next higher infinity than the infinity of the integers; this is called aleph-one. (The next higher infinity after that is aleph-two, and so on). Many mathematicians believe that C and aleph-one are the same, but this cannot be proven. (It is provable that it cannot be proven)
Joseph Louis Lagrange: French mathematician, whom I claim in my thesis is responsible for the first proof of the stability of the solar system. One of Lagrange's more famous books is the Analytical Mechanics, which, he boasted proudly, contains no pictures.
Laplace, Pierre Simon: French mathematician who usually receives credit for proving the stability of the solar system. In my thesis I suggest that it was Lagrange who actually did so. Though I have not been able to demonstrate this yet, I think it was Laplace's writing of Celestial Mechanics, an enormous, five volume tome of celestial mechanics, that established him as the Prince of Celestial Mechanicians. When presented with a copy of some of the initial volumes, Napoleon is said to have remarked, "I see no mention of God in this work". Laplace is said to have replied, "Sir, I have no need of that hypothesis." (In an addition to the story, the tale was related to Lagrange, who added "Ah, but it is such a beautiful hypothesis; it explains a great many things!"
Newton, Isaac: English mathematician, responsible for the development of calculus (though not in its modern form), universal gravitation, and a whole bunch of other things. Some scholars are no longer certain that he was not hit on the head by an apple while watching the Moon rise.