**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!"

- Niccolo Tartaglia, who solved the cubic, failed miserably for the rest of his life (mainly because he spent it trying to discredit Cardano).
- Giralamo Cardano, who published the solution, lived a long unhappy life. His only son was executed for murder; he was later put on trial by the Inquisition for attempting to cast the horoscope of Christ.
- Lodovico Ferrara, who solved the general quartic, was poisoned, probably by his sister, over an inheritance dispute.
- Evariste Galois, who showed the general quintic was unsolvable, died in a duel at the age of 20.
- Niels Henrik Abel, who duplicated and extended Galois' proof independently, finally managed to receive his first faculty position. The notification letter arrived a few days after Abel had died of pneumonia. He was 29.

**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.