This document profiles 12 mathematicians from history and provides 1-4 contributions associated with each one. It includes mathematicians such as Felix Hausdorff, who founded modern topology, and Omar Khayyám, a Persian polymath who wrote on multiple fields including mathematics, astronomy and poetry. Paul Erdős published more papers than any other mathematician and worked on problems in fields like number theory and graph theory.

1. Felix Hausdorff
(November 8, 1868 – January 26, 1942)
He was a German mathematician who is
considered to be one of the founders of
modern topology and who contributed
significantly to set theory, descriptive set
theory, measure theory, function
theory, and functional analysis.
CONTRIBUTIONS:
1. Gromov–Hausdorff convergence
2. Hausdorff paradox
3. Hausdorff distance
4. Hausdorff measure

2. Omar Khayyám
(18 May 1048 – 4 December 1131)
He was a Persian polymath: philosopher,
mathematician, astronomer and poet. He also
wrote treatises on mechanics, geography,
mineralogy, music, and Islamic theology.
CONTRIBUTIONS:
1. Theory parallels
2. Geometric Algebra
3. Binomial Theorem and extractions of root
4. Khayyám-Saccheri quadrilateral

3. Adrien-Marie Legendre
(18 September 1752 – 10 January 1833)
He was a French mathematician. Legendre made
numerous contributions to mathematics. Well-known
and important concepts such as the Legendre
polynomials and Legendre transformation are named
after him.
CONTRIBUTIONS:
1. Associated Legendre polynomials
2. Gauss–Legendre algorithm
3. Legendre's constant
4. Legendre duplication formula

4. Hippocrates of Chios
(470BC – 410 BC)
He was an ancient Greek mathematician. He
was born on the isle of Chios, where he
originally was a merchant. After some
misadventures (he was robbed by either
pirates or fraudulent customs officials) he
went to Athens, possibly for litigation. There
he grew into a leading mathematician.
CONTRIBUTIONS:
1. Stoicheia Elements,
2. Lune of Hippocrates
3. Quadrature of the circle
4. Duplication of the cube

5. Gerolamo Cardano
(24 September 1501 – 21 September 1576)
He was an Italian Renaissance mathematician,
physician, astrologer and gambler. He wrote more
than 200 works on medicine, mathematics,
physics, philosophy, religion, and music. His
gambling led him to formulate elementary rules in
probability, making him one of the founders of the
field.
Contributions:
1. Binomial coefficients
2. Imaginary numbers
3. Quartic equations
4. Cubic equations

6. Johann Heinrich Lambert
(26 August 1728 – 25 September 1777)
He was a Swiss mathematician, physicist,
philosopher and astronomer. He is best known for
proving the Irrationality of π. Asteroid 187
Lambert was named in his honour.
CONTRIBUTIONS:
1. Lambert conformal conic
2. Lambert azimuthal equal area
3. Lambert cylindrical equal area
4. Lambert conical equal area

7. Paul Erdős
(26 March 1913 – 20 September 1996)
He was a Hungarian mathematician. Erdős
published more papers than any other
mathematician in history, working with hundreds
of collaborators. He worked on problems in
combinatorics, graph theory, number theory,
classical analysis, approximation theory, set
theory, and probability theory. He is also known
for his "legendarily eccentric" personality.
CONTRIBUTIONS:
1. Extremal combinatorics
2. Bertrand's postulate
3. Prime number theorem
4. Zero-dimensional

8. Alexis Claude de Clairaut
(3 May 1713 – 17 May 1765)
He was a prominent French
mathematician, astronomer, geophysicist, and
intellectual. His growing popularity in society
hindered his scientific work: "He was focused,"
says Bossut, "with dining and with
evenings, coupled with a lively taste for
women, and seeking to make his pleasures into his
day to day work, he lost rest, health, and finally life
at the age of fifty-two.
CONTRIBUTIONS:
1. Clairaut's theorem
2. Clairaut's equation
3. Clairaut's relation
4. Human computer

9. Marius Sophus Lie
(17 December 1842 – 18 February 1899)
He was a Norwegian mathematician. He largely
created the theory of continuous symmetry and
applied it . Lie's principal tool, and one of his greatest
achievements, was the discovery that continuous Lie
groups could be better understood by "linearizing"
them, and studying the corresponding
generating vector fields. The generators are subject to
a linearized version of the group law, now called
the commutator bracket, and have the structure of
what is today called a Lie algebra.
CONTRIBUTIONS:
1. Lie bracket
2. Lie derivative
3. Lie sphere geometry
4. Lie's third theorem

10. Archytas
(428 BC–347 BC)
He was an Ancient Greek philosopher,
mathematician, astronomer, statesman, and
strategist. He was a scientist of the Pythagorean
school and famous for being the reputed founder of
mathematical mechanics, as well as a good friend
of Plato. He was reputed to have designed and built
the first artificial, self-propelled flying device, a
bird-shaped model propelled by a jet of what was
probably steam, said to have actually flown some
200 meters.
CONTRIBUTIONS:
1. Mathematical mechanics
2. Archytas curve
3. Harmonic mean
4. Doubling the cube