First
sage of Greece,
he founded classical geometry and natural philosophy.
Alchemists have claimed him as one of theirs.
The theorem of Thales
(one of two)
is about two triangles with parallel sides:
The pyramid's shadow is to the pyramid what a man's shadow is to the man
[wow].

In Croton,
he founded the mystic cult of the Phythagoreans,
whose initiated members called themselves mathematikoi.
They are credited with the first proof of the Pythagorean Theorem
(itself known to the Chaldeans
1000 years before). Irrational numbers distressed them...

Founder of metaphysics.
Called the weeping philosopher
(as opposed to Democritus, the laughing philosopher)
Heraclitus argued that all things move and nothing remains still,
which led him to a Mach-like principle of Relativity.

Parmenides upheld the extreme view of
static monism.
He spent some time as a member of the Pythagorean community at Croton.
Zeno was his
eromenos.
At age 65, Parmenides went to Athens and met a youthful Socrates (469-399 BC).

Inventor of rhetoric and borderline charlatan.
His arbitrary explanation of reality with 4 elements (Earth, Air, Fire and Water) and 2 forces (Love and Strife)
dominated Western thought for over two millenia.
Several of his intuitions were correct, though, including the finiteness of the speed of light.

In the most famous of his provocative paradoxes, Zeno asks how
swift-footed
Achilles could overcome a tortoise, since Achilles must first reach the initial
position of the tortoise...
By the time he gets there, the animal is elsewhere and
Achilles is left with a similar challenge, ad infinitum.

A sophist whom
Plato despised
(he portrays him as vain and arrogant, with a wide but shallow knowledge).
Hippias devised the first transcendental curve, known as quadratrix
or trisectrix because the
quadrature of the circle
and the
trisection of an angle
would be trivial if its use was allowed.

The atomists' school in Abdera was founded by his teacher
Leucippus,
himself a student of Zeno and a proponent of the law of causality.
Democritus argued that all was made of indivisible atoms
moving in the void.
One of his followers, the alchemist
Bolus of Mendes,
also signed "Democritus".

Revolutionary founder of Western medicine.
An asclepiad,
said to be a direct descendant (17 or 19 generations) of the legendary
Aesclepius,
Hippocrates studied philosophy under Democritus and learned rudiments of medicine
from his father, Heraclides, and from Herodicus of Selymbria.

A statesman taught by Philolaus (student of Pythagoras)
he taught Eudoxus.
Archytas considered surfaces generated by rotating curves and could
double the cube
by intersecting three of those (defining Archytas' curve in the process).

On a land once owned by someone called Akademos,
Plato created the first institution of higher learning, in 387 BC.
His Academia
lasted 915 years
(Justinian closed it in 529).
Initiation to Geometry was an entrance requirement.
The aim was to teach or discover ideal laws behind appearances.

His definition of the comparison between ratios of (possibly irrational) numbers,
as recorded by Euclid, would inspire the rigorous definition
of real numbers by Dedekind in 1872.
He invented the method
of exhaustion which Archimedes built on.
He was the first Greek scholar to map the stars.

He was the undisputed authority on natural philosophy for two millenia or so.
The lack of discussion of that authority hindered
the development of natural Science more than any other single factor, with
the possible exception of Church doctrine (of which some Aristotelian concepts
were a part).

Father of axiomatic geometry and author of the most
enduring textbook in the history of mathematics: The Elements.
His presentation of the mathematics of his times
would become the centerpiece of mathematical teaching for more than 2000 years.

Copernicus credited him for the idea that Earth rotates on its own axis
and revolves around the Sun.
From rough angular measurements, he estimated the distance to the Sun.
As he couldn't detect the parallax of stars, he declared them to be
extremely distant
(which Archimedes wouldn't accept).

Starting out as a barber, he became an engineer and founded the school
of mathematics at the Library of Alexandria
(he may have served as its first head librarian).
He invented a suction pump, a compressed-air catapult,
a water organ and the
regulated water-clock
(fed by an overflowing vessel).

A native and resident of
Syracuse,
Archimedes studied in Alexandria and maintained
relations with Alexandrian scholars. Although he became famous for designing war
machines, this early physicist was, above all, an
outstanding mathematician.
The 14 Archimedean solids
are uniform.

Apollonius named and studied the
conic sections.
He found that a circle consists of all
points M whose distances to two foci (I,J) are in a fixed ratio
(e.g., 2/3).
He said that planets revolve around the Sun and that the Earth itself might
as well be thought of as moving, like planets do.

Influenced by Ctesibius.
Some of his works were meant to be lecture notes:
Pneumatica
(fluids &
steam)
Metrica (methods and
formulas for areas and volumes, lost until 1896)
Mechanica (statics and simple machines)
Catoptrica (mirrors).
Hero thought that light-rays came from the eyes.

Gaius Plinius Secundus was a public official who wrote a lot.
The 37 books of Historia Naturalis (AD 77)
present, in an anthropocentric way, everything the Romans knew about the natural world.
In this, Pliny cites nearly 4000 authors.

Pedanius Dioscorides was the Greek author of the first major
pharmacopeia
(which never went out of print and remained authoritative for over 1500 years).
The 5 volumes of De Materia Medica (AD 70) present about 600 plants.

A resident of Rome who spent his youth in Alexandria,
he recognized geodesics on a curved surface
as analog to straight lines on a plane.
Shunning arcs of parallels, he defined
spherical triangles as consisting of arcs of great circles.
This was a turning point in
spherical trigonometry.

Claudius Ptolemaeus was a Roman citizen
who wrote in Greek
(his first name may have been Tiberius).
His Almagest dominated astronomy for centuries.
Ptolemy's theorem says that a tetragon is cyclic
iff the product of its diagonals
is the sum of the pairwise products of facing sides.

A Roman citizen
of Greek ethnicity, he started out as physician to the gladiators.
He was so prolific (10 million words) that his surviving works (30%) represent
nearly half of the extant literature of ancient Greece.
His thinking dominated medicine for more than a thousand years.

Earliest female experimentalist on record (signing
Miriam the prophetess, sister of Moses)
she invented the tribikosstill
and the balneum Mariae
(named after her).
F. Hoefer also credits her for
muriatic acid (HCl).
In Alexandria, she reluctantly initiated
Zosimos of Panopolis
(a gentile).

A Diophantine problem is to find an
integer satisfying a polynomial equation
with integer coefficients, or several such equations simultaneously.
Diophantus himself never considered irrational numbers or nonpositive ones.
His age at death was reportedly
x = x/6 + x/12 + x/7 + 5 + x/2 + 4.

Possibly the best mathematician of ancient China, he was a descendant of
Liu Yi, Marquis of Zixiang,
and lived in the state of
Cao Wei (one of the feuding
Three Kingdoms).
He expanded the Jiuzhang Suanshu with his own
commentaries and an appendix which became an official surveying manual.

The theorem of Pappus
(generalized by Pascal in 1639) is a fundamental theorem of
projective geometry.
The name is also used for the two
centroid theorems
published by Paul Guldin (1577-1643) in
Centrobaryca (1635) pertaining to
the surface area and the volume of a solid of revolution.

Daughter of the mathematician
Theon (c. 335-405)
last librarian of Alexandria, who raised her like a boy.
Her teaching of science was seen as pagan.
She was ambushed and skinned alive by a mob of Christian fanatics.
Hypathia's murder marks the beginning of the Dark Ages in the West.

Aryabhata ushered Indian science into a golden age
centered on
Kusumapura and
Ujjain.
His Aryabhatiya (499)
summarized Indian astronomy in 118 verses,
33 of which cover arithmetic, quadratic equations,
spherical and planar
trigonometry,
continued fractions and
power series...

Brahmagupta (the "teacher from
Bhillamal")
was the first to treat 0 like any other number.
Like Diophantus before him,
he pioneered the use of symbols in equations.
He failed to specify that his famous
formula
for the area of a quadrilateral is only valid
for cyclic quadrilaterals.

Abu Musa Jabir ibn Hayyan al Azdi was born in Tus (Persia) but the
Arabs claim him
as one of their own. Geber (or Jabir) made remarkable scientific advances in
practical chemistry but also produced
eponymous gibberish on occult alchemy.

Al-jabr (transposition from one side of an equation to the other) is the technique
which gave algebra its name.
The term is from the title
of the masterpiece published around 810 by
Abu Abdallah Muhammed bin Musa al Khwarizmi.

Key disciple of the three
Banu Musa
brothers, he's best known as Thabit or Thebit.
All later editions of Euclid's Elements were based on his revision.
He was a founder of statics.
He was a Sabian, not a Muslim.

Abu'l-Wafa was the first to build a wall quadrant to observe the stars.
Whenever possible, he determined quantities by giving a ruler
and compass construction for them.
He was an expert in Al-Khwarizmi's "Indian reckoning",
but still wrote out all numbers in arabic letters, for the sake of his audience.

Abu Ali Muhammed ibn al-Hasn ibn al-Haytham al-Basri
was hired by Al-Hakim
and had to feign madness to avoid impossible duties,
until the "Mad Caliph" died (1021).
Early proponent of the scientific method,
Alhazen pioneered optics and anticipated
the first law of Newton (who quoted him).

Celebrated polymath who was first exposed to mathematics by associating with
Abu Nasr Mansur (970-1036)
of sine law fame.
Al-Biruni pioneered scientific methods in astronomy and geology.
First mathematician to point out the limited validity
of Brahmagupta's simplified formula.

The word Khayyam means "tentmaker" (possibly, his father's trade).
His
Rubáiyát
("quatrains") were translated in 1859 by
FitzGerald.
Khayyam reformed the calendar of
the Seljuq empire (1079).
He solved cubic equations with
conic sections, stating that
ruler and compass didn't suffice.

Last and greatest mathematician in the Golden Age of Indian mathematics.
He developed trigonometry for its own sake, including spherical trigonometry,
and introduced the addition formula:
sin (x+y) = sin x cos y + cos x sin y
He conceived derivatives
and stated Rolle's theorem.

Educated at Oxford University,
of which he became Chancellor in 1215 (until 1221).
Grosseteste introduced the earliest teaching of the
scientific method in Oxford
(comparing theories with observations). After holding other ecclesiastical posts, he became
Bishop of Lincoln in 1235.

He ended a mathematical lull of eight centuries in the West.
As a teenager in Algeria, Fibonacci learned the Hindu-Arabic
decimal system
that he would advocate in Europe.
In Liber Abaci (1202) he discussed many
computational puzzles,
including one
about the Fibonacci sequence...

Nicknamed Doctor Mirabilis. He went to the University of Paris to take a degree
(1241) and he started lecturing on Aristotle there (1234-1247) before returning
to Oxford. Influenced by Grosseteste, Roger Bacon became the
most active early proponent of the scientific method in Europe.

Arguably, the foremost Medieval logician.
His enduring contribution to natural philosophy is the "principle
of parsimony" known as Occam's Razor
(the simplest explanation compatible with observations is preferred).

In 1327 and 1340, Joannes Buridanus was rector of
Paris
where he had studied under Ockham
(whom he condemned in 1340).
Buridan seeded Copernican ideas. He contributed to
probabilities and optics. His concept of
impetus (c.1340)
anticipated momentum.
Excommunicated for
nominalism.

Star student of Jean Buridan,
Nicolas Oresme is credited with the introduction of
fractional exponents and the graphing of functions.
He also established the
divergence of the harmonic series.
Oresme anticipated analytic geometry, the law
of free fall and chemical structures...

Madhava gave the first examples of
power series
(besides geometric series)
as expansions of trigonometric functions (sin, cos, arctg).
Madhava's knowledge was perpetuated and expanded by the school he founded in Kerala
and may have influenced similar developments in the West, much later.

Mathematical prodigy, earliest publisher of printed scientific works.
Johannes Müller von
Königsberg
signed Joannes de Monte Regio.
( "Regiomontanus" was coined in 1534,
by Melanchthon).
Cardano scorned him for publishing
Jabir ibn Aflah's
spherical trigonometry without proper credit.

A stellar Renaissance painter,
he left 13000 pages of illustrated notes on science and engineering
(in mirror-image cursive). He was taught mathematics by
Luca Pacioli (1447-1517)
as he lived with him in Milan, illustrating
"De divina proportione" (1496-1498)
with skeletonic polyhedra.

Philippus Aureolus Theophrastus Bombastus von Hohenheim chose the pseudonym
Paracelsus in honor of the encyclopedist
Celsus.
He is the first systematic botanist.
He named zinc (1526)
and revolutonized medicine (without freeing it from superstition) by using
mineral chemicals.

First scholar to use negative numbers routinely.
In 1545, he revealed the solution of cubic equations obtained by
del Ferro (1465-1526)
in 1516 and rediscovered (1535-02-13) by
Tartaglia
(1500-1557).
It had been extended to quartics, in 1540, by his own assistant
Lodovico Ferrari
(1522-1565).

Ambroise Paré was a royal military surgeon.
On one occasion on the battlefield, he had to use a makeshift ointment.
He observed that the soldiers so treated recovered much better than those
who underwent the formerly "recommended" treatment (i.e., burning wounds with oil).

Breaking with the precepts of Galen,
Andreas Vesalius Bruxellensis revolutionized medicine in 1543
with the first modern book on
human anatomy, based on the detailed observations he made during
the dissections that he carried out in front of medical students
at the University of Padua.

His name is also spelled Viette
(latin: Franciscus Vieta).
Viète pioneered modern algebraic notations,
where known constants and unknown quantities are represented by letters.
The trigonometric
law of tangents (c. 1580)
is due to him.
In 1593, he gave an expression of p
as an infinite product.

Tyge Ottesen Brahe was from the high Danish nobility.
His Uraniborg observatory,
on Hven island,
cost 1% of the state budget but allowed precise (naked-eye)
observations of planetary positions which made possible the work of
Kepler.

Flemish engineer who introduced decimal fractions (1583) shortly after
Viète (1579).
Stevin wrote in Dutch and coined many Dutch scientific terms
(without the Latin/Greek roots used in other languages).
He found that the pressure exerted by a liquid at rest in a vessel depends only on depth (1586).

Known as Neper to the French, he invented an
early version of logarithms
which he spent years tabulating.
This improved upon prosthaphaeresis
(multiplication using trigonometry).
Common (decimal) logarithms are due to his younger contemporary
Henry Briggs (1561-1630).

Using his own pulse as a timer,
Galileo discovered the
pendulum isochronism in 1581.
He found that all bodies fall with the same acceleration and
declared mechanical laws valid for all observers in uniform motion.
He made the first telescopic observations of celestial bodies.

Kepler's precise calculations helped establish heliocentric
astronomy. In 1609 and 1619,
he published his famous 3 laws of planetary motion.
He studied optics,
polyhedra,
logarithms, etc.
Arguably,
he paved the road to Calculus.

William Harvey started modern experimental medicine with his discovery
of the circulation of the blood.
He had been a student at Padua,
where the Flemish anatomist Andreas Vesalius (1514-1564)
had started encouraging students to observe
rather than conform to the precepts of Galen.

Building on the fundamental results of Pappus,
Desargues invented
projective geometry in 1639.
That innovation was largely ignored, except by the likes of Pascal
and La Hire,
until a key manuscript rediscovered in 1845 was published in 1864,
following a remarkable rebirth of the subject.

Descartes attended the famous Jesuit college of
La Flèche
from 1607 to 1615. He met his scientific mentor
Isaac Beeckman (1588-1637)
in 1618. He introduced cartesian geometry in one of the three appendices
to Discours
sur la méthode (1637).

In Pisa, Cavalieri was mentored by
Benedetto Castelli (1578-1643)
who put him in touch with Galileo.
Cavalieri's principles
can be construed as the preliminary conceptual foundations for integral calculus,
stating (in modern terms) that the
integrals of equal functions are equal...

Orphan. Assistant to Castelli, then Galileo.
Torricelli invented the barometer in 1644: He
figured out that what's above a column of 760 mm of mercury is a near-perfect vacuum
(just rarefied mercury vapor). What pushes the liquid up the tube
is the (variable) atmospheric pressure.

Appointed to the Savilian
Chair of Geometry at Oxford by
Oliver Cromwell
in 1649, John Wallis held that position for more than 50 years.
In 1655, he published his great Arithmetica Infinitorum,
which helped pave the way for the introduction of modern Calculus
by Newton and Leibniz.

At 16, he generalized
the theorem of Pappus. At 19, he built a celebrated
mechanical calculator.
In 1647, Pascal thought of using a Torricelli barometer as
an altimeter,
which established experimentally (1648) the origin of atmospheric pressure.
The SI unit of pressure (Pa) is named after him.

The Japanese Newton.
Second son of a Samurai warrior,
he was adopted by a technocrat (Gorozaemon SEKI )
whose name he took.
Some of Seki's discoveries predate their Western counterparts:
Determinants (1683)
Bernoulli numbers, etc.
He taught Katahiro TAKEBE (1664-1739).

Earliest mathematician in a
family that would produce many
(but none among his descendants).
With his younger brother Johann,
Jacob pioneered the calculus of variations
(which Euler would tackle in 1744).
He found Bernoulli numbers (independently of Seki)
and formalized probability theory.

Father of Daniel and main teacher of Leonhard Euler.
Initiated by his older brother Jacob, he collaborated with him
on early topics in the calculus of variations.
Hired to teach Guillaume de l'Hôpital, Johann had to
name after his student the famous rule he discovered during that work-for-hire.

Pierre-Louis Moreau de Maupertuis used his
principle of least action (1744)
to reformulate Newtonian mechanics.
This paved the way for Lagrangian
and Hamiltonian mechanics and provided an elegant key for
an historical derivation of
Schrödinger's equation, published in 1928.

At the same time as
Watson
(1746) Franklin
formulated the law of conservation
of charge by positing opposite signs for
resinous (-) and vitreous (+)
electricity.

At 19, Gabrielle-Emilie de Breteuil married the Marquis
Florent-Claude du Chastellet.
She was the lover of Voltaire whom she
and her husband protected in their château.
She was tutored by Maupertuis (1733) and
Clairaut (1735). She
popularized
the concept of energy
introduced by Leibniz.

He solved the Basel Problem in 1735.
The most prolific mathematician of all times,
Euler became totally blind in 1771. He still produced nearly half of his 866 works after 1766
(in St. Petersburg)
with the help of several assistants, including
Nicolaus Fuss
(1755-1826) who joined in 1773.

Gabriele Manfredi
(1681-1761)
initiated her to higher mathematics and newtonian physics.
In 1732 (at age 21) Laura Bassi
became the second woman
to earn a doctorate and the first to teach at a European university
(Bologna).
She was finally named professor of physics there, in 1776.

At age 16, he introduced the study of space curves.
He was the youngest member ever of the
Académie des Sciences (July 1731).
Clairaut's theorem
(1740) says that, provided it's continuous,
a partial derivative with respect to several variables doesn't depend on the order of the differentiations.

Child prodigy and author of the first mathematical book by a woman (1748).
In 1750, she was appointed to the chair of mathematics at Bologna by
Pope Benedict XIV
but she never went there (the first woman to hold
a chair in Europe was thus Laura Bassi, in 1776).

His 6-volume mathematical textbook (1770-1782) was once standard for students
wishing to enter Polytechnique
(this was also used at
Harvard
for calculus).
His research was focused on the theory of equations.
Bézout's
little theorem says that the polynomial
P(x)-P(a) is divisible by (x-a).

Antoine Lavoisier founded quantitative chemistry by establishing that
mass is conserved in any chemical transformation.
He was infamously executed during the French Revolution because of his
rôle as a tax collector.

Correctly interpreting the 1791 observation by
Luigi Galvani (1737-1798) of muscle contractions
in a dead frog, Volta reasoned that electricty is generated upon contact of two different metals.
Replacing living tissue by paper soaked with saline electrolyte, he built the first battery in 1799.

In 1768, he succeeded his mentor
Charles
Bossut to the chair of mathematics at the
Ecole
de Mézières.
Monge would use that school as a model for
Ecole Polytechnique, founded in 1794 with himself as Director and
instructor in descriptive
geometry (the drafting technique he had devised in 1765).

Initiated to mathematics, in
Caen, by
Christophe
Gadbled
and Pierre Le Canu,
Laplace was mentored by d'Alembert (in Paris)
and became one of the most influential scientists ever
(Laplacian,
Laplace transform).
With Lavoisier, he proved respiration
to be a form of combustion (1783).

Before Jenner, risky variolation and other inocculations were
believed to induce immunity to dangerous diseases
(20% of human deaths were due to smallpox).
Putting some human lives at risk, Jenner proved that innoculation with
harmless cowpox did protect against the dreaded smallpox.

Legendre was one of the greatest contributors to the mathematics of his times.
Many concepts are named after him.
At left is what seems to be
his only extant portrait
(it was found among 73 caricatures of members of the French academy of Sciences).

In 1776, under Monge,
Meusnier read to the Académie
two papers about surface curvature and the
helicoid
(both published in 1785).
With Lavoisier, he mass-produced hydrogen
by oxydizing
600°C iron with water vapor (1777).
Meusnier fought as a
general
and died in battle near
Mainz.

Joseph Nicéphore Niépce invented photography (1826).
He built the first internal combustion engine
(Pyréolophore, 1807) with his brother
Claude (1763-1828).
His cousin Abel Niépce de Saint-Victor
(1805-1870) would photograph radioactivity in 1857 (39 years before Henri Becquerel did).

In January 1795, Jean-Baptiste Joseph Fourier was the star trainee in the new
Ecole normale de l'an III (the forerunner of
ENS)
simultaneously teaching at Polytechnique.
He is the founder of
Harmonic Analysis
(cf. Fourier transform).

Notorious for his
two-slit experiment
demonstrating the wavelike nature of light (1802)
and for Young's modulus of elasticity (1807).
Young's rule gives the posology for an n-year old child as
n/(n+12) of the adult dose.
Young paved the way for the decoding of hieroglyphics by
Champollion.

Appointed professor of mathematics at Polytechnique in 1809.
In september 1820, he discovered that
like currents attract each other whereas opposite currents repel.
The effect is now used to define the SI unit of current, which is named after him.

At 13, the story of the death of Archimedes
inspired her to become a mathematician.
She was 18 when Polytechnique opened
(it was male-only until 1972)
and made available Lagrange's lecture notes.
This gave her a start to correspond with him and others
(signing Monsieur LeBlanc at first).

At the age of 7, the
Prince of Mathematics found instantly the sum (5050) of all integers
from 1 to 100 (as the sum of 50 pairs, each adding up to 101).
At age 19, his breakthrough about
constructible polygons helped him choose
a mathematical career.

Among his many mathematical contributions is a very abstract construct in
analytical mechanics (Poisson
Brackets, 1809) which helped Dirac
formulate a precise correspondence between classical and quantum
mechanics (Sunday, Sept. 20, 1925).

He taught analysis and geometry at Polytechnique from 1810 to 1830,
at the peak of his creativity (electromagnet, 1820).
A popular left-wing deputy elected in 1830, Arago became Minister of Marine and War in 1948 and
was instrumental in abolishing slavery in the French Colonies (1848).

In 1814, his observation of the Sun's dark-line spectrum (Fraunhofer lines) marked the
beginning of astrophysics.
Fraunhofer is also remembered for related studies of diffraction in optical systems with small
Fresnel numbers (Fraunhofer diffraction).
Knighted in 1824 (Bavaria).

Trained in
Caen (1801-1804) then at Polytechnique.
Poor physicist at first...
In 1821,
Augustin Fresnel established (with Arago)
that light is a transverse wave
whose two polarizations don't interfere with each other.
He invented
Fresnel lenses for use in lighthouses.

POW in Russia for 15 months (1812-1814) he brought back
from Saratov
the 7 notebooks in which he had invented
modern projective geometry.
Promoted to Colonel in 1845 and General in 1848,
Poncelet headed Polytechnique from 1848 to 1850.

A devout royalist, Cauchy wrote 789 papers in all areas of the mathematics and
theoretical physics of his time. In 1821, his Cours d'analyse
at Polytechnique
made analysis rigorous.
He originated the calculus of residues (1826) and
complex analysis (1829).

In 1831, Faraday discovered the
Law of Electromagnetic Induction, which
made the electric era possible.
He is widely regarded as one of the greatest
experimentalists who ever lived.
Yet, he had little or no grasp of higher mathematics.

He was Lucasian Professor
(1828-1839) at Cambridge but never taught.
He designed two computing machines:
The Difference Engine (funded in 1822) was never completed.
The more advanced Analytical Engine
would have been the first true computer (Ada Lovelace wrote programs for it).

He gave the terms
work
(travail) and kinetic energy
their precise mechanical meanings.
At Ponts-et-Chaussées since 1832,
Coriolis inherited the chair of Mechanics there, in 1836, upon the death of
Navier,
and became director of studies at Polytechnique.

Professeur of geodesy at Polytechnique from 1841 to 1851,
he inaugurated the Sorbonne
chair of projective geometry,
then called higher geometry (1846-1867).
His reputation as a science historian was all but ruined when he
bought forged manuscripts (1861-1869)
from Denis Vrain-Lucas.

In his first published paper (1826) he devised geometrical
inversion
(paving the way for homographic transforms)
which embodies duality for polyhedra,
convexes, etc.
He was one of the architects of the rebirth of
projective geometry in the first half of the eighteenth century.

Niels Abel
produced many brilliant results during a short life spent in poverty:
Non-solvability of quintic equations by radicals,
double periodicity of the elliptic functions, etc.
An offer for his first professorship
(at Berlin)
arrived two days after he had succombed to tuberculosis.

An inspiring teacher, he was an outstanding and prolific creator of mathematics
who has been likened to Euler.
He introduced ¶ and
Jacobians in 1841.
Jacobi admired
Poisson brackets
and proved that they satisfy what's now called
Jacobi's identity.

Johann Peter Gustav Lejeune-Dirichlet.
signed Gustav Lejeune Dirichlet, (no hyphen)
published as P.G.L. Dirichlet
and was quoted as Lejeune-Dirichlet. He contributed to
number theory, mechanics and
analysis.
He was the first to consider unrestricted functions.

A calculating prodigy who lost to
Zerah Colburn
at age 8, Hamilton started to teach himself higher mathematics at 13.
In 1833, he devised a version of
rational mechanics
(based on conjugate momenta) which would help clarify
quantum mechanics later.
He invented quaternions in 1843.

Against strong religious animosity (which lasts to this day in the US)
Darwin established that the mechanism of natural selection
was powerful enough to explain the evolution of the humblest ancient lifeforms
into the most advanced modern ones, featuring very sophisticated organs.

Many of Liouville's 400+ papers include key contributions, like his
conservation
of Hamiltonian phase-measure. In 1836, he founded the
Journal de mathématiques pures et appliquées and promoted
the work of others, including the late Evariste Galois.

Around 1832,
he pioneered the modern approach to vectors
and went on to invent exterior algebra (the correct basis
for Cartan's differential forms
and/or Bourbaki's
"Stokes' theorem").
Grassmann had little mathematical influence during his own lifetime
(he became successful as a linguist).

Galois theory is about symmetries of polynomials on
fields. Galois "didn't have time" to
extend that to transcendental functions (nobody else has done so).
He died in a stupid duel at the age of 20 and his
fundamental work might have been lost if Liouville hadn't
revived it in 1843.

He introduced the notion of higher-dimensional vectors
(between 1850 and 1852, full treatise published in 1901).
He pioneered multi-dimensional Riemannian manifolds by considering the
3D-hypersurface of an 4D-hypersphere.
Schläfli also classified
allregular polytopes.

In 1838, he founded the preparatory school at Sainte-Barbe
with Sturm and Liouville.
His left-wing activism damaged his academic career.
He was elected to the French National Assembly.
Catalan's conjecture (1843)
saying that the only solution of
1+x^{m }= y^{n} is 1+2^{3 }= 3^{2},
was proved in 2002.

The father of analysis
spent 15 years teaching secondary school before one paper
earned him an honorary doctorate and a professorship.
He gave the rigorous
metric definition of limits and invented the
concept of analytic continuation.

Daughter and heiress of Lord Byron (the poet) whom she never knew.
Ada was introduced by
Mary Somerville to
Charles Babbage on June 5, 1833.
She then developped an intense interest in the mathematics of computation
and is now regarded as the first computer programmer.
[ Video ]

Home-schooled Russian aristocrat. His mathematics tutor was the textbook author
Platon Nikolaevich Pogorelski (1800-1852).
Chebyshev contributed to number theory, algebra, analysis, mechanics, etc.
In 1850, he derived
Bertrand's postulate
from the
totient function's asymptotics.

He wrote 996 papers on many mathematical subjects
(200 of these while praticing law, for 14 years).
In 1858, Cayley established (without a formal proof)
the Cayley-Hamilton theorem:
A matrix is a zero of its characteristic polynomial.

We use his initial (H) for
enthalpy, not for the
Helmholtz free energy (F).
Helmholtz is primarily known for his work in physics (thermodynamics, acoustics,
elasticity, etc.)
but the fundamental theorem of vector calculus (3D only)
is also named in his honor
(Helmholtz decomposition).

After one year at Polytechnique, the military management
dismissed him because of a congenitally deformed right leg.
Returning as a teacher, five years later, he contributed to number theory,
orthogonal polynomials and elliptic functions.
He proved e transcendental in 1873.

Louis Pasteur was a trained chemist who separated chiral isomers
by sorting the different crystals they produce.
He proved the germ theory of infectious diseases
and invented pasteurization.
Motto: Fortune favors the prepared mind.

Famous for his credo "God made the
natural numbers;
all else is the work of man", Kronecker
championed constructivism. He strongly opposed his former
student Georg Cantor and the
emerging nonconstructive
Set Theory.

Born William Thomson, Lord Kelvin was knighted
in 1866 and raised to the peerage in 1892 (Baron Kelvin of Largs).
The SI unit of temperature is named after this
mathematician noted for his engineering work (e.g., transatlantic telegraph).

Applying Pasteur's ideas, he introduced antiseptic surgery while working
at the Glasgow Royal Infirmary.
Lister used carbolic acid (phenol)
to sterilize instruments and clean wounds.
This reduced post-operative infections and made surgery safer.
Baronet in 1883, he became a Baron in 1897.

In 1864, he devised Maxwell's equations
which unify electricity and magnetism, by describing electromagnetic
fields traveling at the speed of light.
In 1866, Maxwell proposed (independently of
Boltzmann) the Maxwell-Boltzmann
kinetic theory of gases.

Julius Wilhelm Richard Dedekind was
the last doctoral student of Gauss
(1852)
but he also learned much from Dirichlet
after his doctorate. On 24 November 1858, he defined every real number
as a Dedekind cut
of rationals. In 1871, he introduced algebraic
ideals.

In 1869,
he presented a classification of chemical elements
(based mostly on atomic masses) which showed periodic patterns in their chemical properties.
He predicted the properties of 3 unknown elements which were discovered shortly thereafter:
Ga (1871), Sc (1879) and Ge (1886).

Bringing to a great conclusion the works of
Gauss,
Bolyai,
Lobachevski and
Riemann
on non-Euclidean geometry, he showed that geodesics matched straight lines on the plane only
for surfaces of constant
curvature. His pseudosphere
(generated by rotating a tractrix) is the key example (1868).

A universal mathematician and one of the greatest teachers of the 19-th century,
he inspired Lie, Klein,
Borel and Lebesgue.
He invented the topological concept of
homotopy (1866).
Camille Jordan was appointed professor of Analysis at Polytechnique in 1876.

Mach would only consider relative motion between objects, irrespective of
absolute Newtonian space.
He studied the shockwaves produced by fast projectiles
(the Mach number of a projectile is the ratio of its speed
to the speed of sound in the surrounding fluid).
Mach was Pauli's godfather.

Founder of modern optics.
His industrial commitments to the instrument-maker
Carl Zeiss (1816+1888) and
the glassmaker Otto Schott (1851-1935)
prevented Abbe from accepting a professorship at
Berlin
(offered by Helmholtz).

He tied his definition of integrals (1870) to that of
Riemann in 1875.
The Darboux formulas define the normal
and geodesic curvatures as well as the geodesic torsion for a curve drawn on a surface.
He was a biographer of Poincaré.
Darboux was elected to the Académie des Sciences in 1884.

He's the man who explained why the sky is blue
(Rayleigh scattering).
He described surface acoustic waves (SAW
or Rayleigh waves, 1885) before they were observed in earthquakes.
He earned the Nobel prize (1904) for his
1892
discovery of Argon.
Rayleigh was J.J. Thompson's advisor.

With Felix Klein,
Sophus Lie originated the investigation of the continuous
groups of symmetry now named after him.
The study of Lie groups and the related
Lie algebras would become a major branch of
20-th century mathematics, with applications to
quantum mechanics.

A proponent of atomic theory and the father of
statistical physics. We call
Boltzmann's constant
the coefficient of proportionality between entropy
(in J/K) and the natural logarithm of
the number W
of allowed physical states.

Cantor's diagonal argument shows that
the points of a line are not countable.
More generally,
Cantor's Theorem
states that no function from a set to its powerset
can possibly be surjective,
which establishes an infinite sequence of increasing
infinities.

The most successfull
inventor ever.
His 1093 US patents cover the phonograph, light-bulb, motion picture camera...
In 1876, he created the first industrial research laboratory at
Menlo Park, NJ.
He favored DC current, which lost out to Tesla's
AC generation and distribution of electric power.

Investigating Lie groups independently of Lie and Klein,
he fully classified simple Lie groups in 1887
(as confirmed by Cartan in 1894):
5 exceptional Lie groups
(E_{6 },
E_{7 },
E_{8 },
G_{2 },
F_{4 })
and three regular families:
special linear groups SL(n),
orthogonal groups O(n), symplectic groups Sp(2n).

Born on 1849-4-25 (43^{2}, 2^{2}, 5^{2 })
to a Prussian government official, he married the granddaughter
of Hegel in 1875.
The noncyclic group of order 4 bears his name.
As first president of the
ICMI (1908) he was instrumental in bringing
Calculus (back) to secondary schools worldwide.

Sofia Vasilyevna Kovalevskaya was born Sonya Korvin-Krukovskaya.
Weierstrass tutored her privately (1870-1874) and helped her
become the first female professor at a European university (Stockholm, 1889)
since the days of Laura Bassi (1776) or
Maria-Gaëtana Agnesi.

In 1884, he started the investigations of quadratic differential forms which led him
to invent tensor calculus
(1884-1894). The text he published about that with
Tullio Levi-Civita
in 1900 would enable Einstein to formulate
General Relativity in 1915.

Among the many contributions of H.A. Lorentz is
the coordinate transformation
which is the cornerstone of Special Relativity.
In 1892, Lorentz proposed a
theory of the
electron (discovered by Perrin in 1895 and
J.J. Thomson in 1898).

Doctoral student of Hermite
(1879)
and last universal genius. Quintessential
absent-minded professor (cf. Savant Cosinuscomic strip).
Poincaré conceived Special Relativity
before Einstein did. His mathematical legacy includes
chaos theory and contributions to topology.

At least 272 patents
were awarded to Tesla in 25 countries.
His work is the basis of modern alternating current
(AC) electric power distribution.
In 1960, the
SI unit of
magnetic induction (magnetic flux density) was named after him.

In 1887, Heinrich Rudolf Hertz
discovered the photoelectric effect, whose
explanation by Einstein, in 1905, would establish the existence
of photons.
In 1888, he made the first transmission of a signal by
radio waves.
The SI unit of frequency
(symbol Hz) was named after him, in 1960.

Planck combined the formulas of
Wien (UV) and
Rayleigh (IR) into
a unified expression for the
blackbody spectrum.
On Dec. 14, 1900, he justified it by proposing that exchanges of
energy only occur in discrete lumps,
dubbed quanta.

One of the most powerful mathematicians ever, David Hilbert gave a famous
list of 23 unsolved problems in 1900. Quantum Theory
is formally based on the complex normed vector spaces
which are named after him.

Pioneering convex geometry, he proved
an early version of the separation theorem (of Hahn-Banach)
and called A+B the set of all sums with one addend
in A and the other in B.
The triangular inequality
for the L^{p}norm (1896) and
the relativisticscalar product (1908)
are named after him.

In 1892, he obtained his doctorate and was awarded the French Academy's Grand Prix
for completing the work of Riemann on the Zeta function.
He authored one of the first two proofs of the Prime number theorem in 1896.
He gave functional analysis its name in 1910.
Deeply influential.

Madame Curie (née Maria Salomea Sklodowska )
was the first woman to earn a Nobel prize and the first person to earn two.
In 1898, she isolated two new elements (polonium and radium)
by tracking their ionizing radiation, using the electrometer
of Jacques and Pierre Curie
(her husband).

In 1908, Henrietta Swan Leavitt
published the period-luminosity relationship for Cepheid variable stars,
which reveals their actual distances, even when stellar parallax is undetectable.
This paved the way for the first measurement of the expansion of the Universe
by Edwin Hubble (1929).

In a Hausdorff space (1914) two distinct points
are always disconnected.
In 1919, he introduced fractional dimensions
and defined d-dimensional measures.
Hausdorff published literary work as Paul Mongré.
Unable to escape the Nazis, he committed suicide with his wife and sister-in-law.

In 1913, Cartan established, from a purely geometrical standpoint, the relations that
lead to the quantization of spin.
He developed exterior calculus
and published his Theory of Spinors as a textbook
in 1935. Godfather of Bourbaki and father of
key bourbakist Henri Cartan
(1904-2008).

A Sainte-Barbe bursar, he ranked first in the top three
French academic competitions of 1889:
Concours Général,
Polytechnique, Ecole Normale.
He chose to enter the latter.
Borel developed point-set topology and
founded Measure theory.
Elected to the Académie des Sciences in 1921.

British physicist born in Nelson, New Zealand.
His investigations of alpha and beta decay (which he so named) earned him
a Nobel prize before he moved to
Manchester, where he
supervised the Geiger-Marsden
experiment (1909) and inferred the planetary model of the atom (1911).

Building on the work of Jordan and
Borel (his advisor) he laid the goundwork of
measure theory in 1901 and went on to revolutionize the notion of
definite integration in his doctoral dissertation (1902).
Lebesgue was elected to the Académie des Sciences on 29 May 1922.

Known only by his initials G.H. (for Godfrey Harold)
Hardy was asexual, entirely devoted to mathematics and cricket
(a nonpractising homosexual, said Littlewood).
His collaboration with Littlewood is legendary. So is the way Hardy
recognized and guided Ramanujan's raw genius.

A student of Ludwig Boltzmann, she became a collaborator
of Otto Hahn who was awarded a
Nobel prize (1944)
for their joint work.
With Otto Frisch (her nephew) Lise Meitner gave
nuclear fission its name (Kernspaltung).
She correctly explained the related
mass defect (1938).

Early in his career, Bertus Brouwer
founded modern topology.
He later championed the incompatible philosophy of
intuitionism
which considers only sets whose elements can be shown to belong in finitely many steps.
As topology isn't intuitionistic, he wouldn't teach it at all!

Emmy Noether discovered the remarkable equivalence between symmetries in physical laws
and conserved physical quantities
(Noether's theorem, 1915).
Her considerable legacy also includes
three Isomorphism Theorems named after her (1927).

Littlewood had
22 doctoral students
but, like Hardy, never bothered to take a doctoral degree himself.
In 1910 or 1911, he started a prolific collaboration with
G.H. Hardy which spanned 35 years.
He was so discreet that rumors once circulated that he was just a
figment of Hardy's imagination.

In 1913, Bohr started the quantum revolution
with a model where
the orbital angular momentum
of an electron only has discrete values.
He later spearheaded the Copenhagen interpretation
(i.e., probabilistic measuremments cause the collapse of otherwise
linearly-evolving quantum states).

In 1908, Weyl obtained his doctorate from
Göttingen
under Hilbert.
He was enthralled by symmetry in mathematical physics.
In 1913, Weyl became a
colleague of Einstein's at the
ETH Zürich.
He befriended Schrödinger in 1921.
Weyl introduced compact groups (1923-1938).

In 1926, Schrödinger matched observed quantum behavior with the properties of
a continuous nonrelativistic wave obeying the
Schrödinger Equation.
In 1935, he challenged Bohr's Copenhagen Interpretation,
with the famous tale of Schrödinger's cat.
He lived in Dublin from 1939 to 1955.

Ramanujan lacked a formal mathematical education but, in 1913, a few of his early results
managed to startle G.H. Hardy (1877-1947) and
J.E. Littlewood (1885-1977) who invited him to
Cambridge in 1914.
Ramanujan has left an unusual legacy of brilliant unconventional results.

In 1923, he proposed that any particle could behave
like a wave of
wavelength inversely proportional to its momentum
(this helps justify Schrödinger's equation).
He predicted interferences for an electron beam hitting a crystal.

Born in Austria, he fled the Nazis in 1937.
Artin is credited for modernizing
Galois theory.
He passed on his interest in
subassociativity to his student
Max Zorn (of lemma fame).
Emil Artin was also the doctoral advisor of two Bourbakists:
John Tate (Abel prize, 2010) and Serge Lang.

In 1925, Wolfgang Pauli formulated the exclusion principle
which explains the entire table of elements.
His Godfather was Ernst Mach.
Pauli's sharp tongue was legendary; he once said about a bad paper:
"This isn't right; this isn't even wrong."

In 1926, Fermi helped formulate the Fermi-Dirac statistics
obeyed by what we now call fermions.
He identified the neutrino in beta-decay.
He discovered slow neutrons and the radioactivity they induce.
On December 2, 1942, Fermi produced the first self-sustaining nuclear
chain reaction.

In 1925, Werner Heisenberg replaced Bohr's semi-classical orbits
by a new quantum logic which became known as
matrix mechanics (with
the help of Born and Jordan).
The relevant noncommutativity entails his
uncertainty principle.

In 1925, Paul Dirac came up with the formalism
on which quantum mechanics is now based.
In 1928, he discovered a relativistic wave function for the electron,
predicting the existence of antimatter (observed by
Anderson in 1932).

He constructed functions whose Fourier series diverge
almost everywhere (1922)
or everywhere (1926).
In 1933, he laid the foundations of axiomatic probability theory.
Based on his 1954 work, the long-term stability of the
solar system can almost be established
(KAM theorem).

He is credited with the
stored program architecture (1946) whereby a computer uses
its primary memory space to store both the data it operates on and the
codes for the programs it executes.
Von Neumann also pioneered game theory and
decision analysis.

Son of Elie Cartan (1869-1951).
A key founder, with Weil, of the
Bourbaki group (1935) which would consume a
large part of his research activities.
He was the leading professor at ENS for several decades.
Cartan was instrumental in reconciling French and German mahematics after WWII.

In 1944, Thomas Harold Flowers built the first large-scale electronic
computer (Colossus) at
Bletchley Park.
As the accomplishment remained classified for decades, Flowers was deprived
of the glory which went instead to
Mauchly and
Eckert for the
ENIAC (Philadelphia, 1946).

The completeness theorem in his dissertation (1929)
states that a statement true in everymodel of an axiomatic
system is provable in it. His more famous incompleteness theorem
(1931)
says that, in any model of a set of axioms covering arithmetic,
some true statements are not provable.

Older brother of the philosopher Simone
Weil (unrelated to the politician
Simone Veil)
he was the leading founder of Bourbaki.
Weil created algebraic geometry
and, arguably, charted the course of much abstract mathematics
in the twentieth century.

Arguably the most brilliant of a dozen
Via Panisperna boys
selected by Fermi (including the likes of
Wick and
Segrè).
Majorana published only 9 scientific papers but left a huge 10,000-page legacy
of notebooks written between 1927 and 1932.
He organized his own disappearance in March 1938.

Harold Scott MacDonald Coxeter was a British-born Canadian mathematician
teaching at Toronto.
He put forth
reflection groups. He wrote
Introduction to Geometry (1961) and
Regular Polytopes (1963).
A correspondant of Martin Gardner, he inspired
Bucky Fuller and
M.C. Escher.

Top code-breaker of Bletchley Park.
A Turing Machine is a finite automaton endowed with an infinite
read/write tape on which it can move back and forth, one step at a time.
Turing showed that this type of machine is actually capable
of computing anything that any other machine could.

Paul Erdös wrote over 1500 papers with 511 collaborators.
He contributed many conjectures and proved some great ones.
Faced with antisemitism, he left Hungary in 1934 and spent the
rest of his frugal life on the road, touring mathematical centers.

"One night in 1944", he figured out that the distributions
used in theoretical physics
(including Dirac's delta) weren't pointwise functions
but linear forms over a restricted set of
smooth test functions. The Fourier transform turns out to
be a linear automorphism
among tempered distributions.

In 1949, he introduced
Feynman diagrams
to describe the relativistic quantum theory of
electromagnetic interactions known as
Quantum
electrodynamics (QED).
This has helped visualize all other types of fundamental interactions ever since.

Robinson's non-standard analysis (1961)
gave a rigorous footing to the infinitesimals
introduced by Leibniz (1675)
thus providing an alternative basis for analysis
(competing with the approach made standard by Cauchy
in 1821).
This was an early application of
Model theory.

The notion of a
Nash equilibrium
(in his 1950 dissertation about non-cooperative games)
makes game theory relevant
to many real-life situations. Nash battled
schizophrenia for decades,
but willed it off before receiving
the Nobel prize in economics, at 66, and the
Abel prize, at 86 (2015).

John Stewart Bell earned his PhD in nuclear physics at
Birmingham in 1956.
In 1960, he and his wife
(Mary Ross)
gave up tenured positions to work at
CERN for the rest of their careers.
After a year-long sabbatical from CERN, John published his masterpiece:
"On the EPR paradox" (1964).

In 1967, he formulated the electroweak unification of the
weak nuclear force and electromagnetism,
predicting a massive neutral messenger
particle (the Z boson) which was first observed in 1979.
Steven Weinberg gave the Standard Model its name.
["To Explain the World", 2015.]

His invention of the technique of forcing revolutionized logic
and allowed him to prove, in 1963, the undecidability of Cantor's
continuum hypothesis (CH):
Gödel had shown CH to be compatible with the axioms of set theory and
Cohen proved the same for its negation...

In 1970, Conway found the simple rules of a cellular automaton
(the Game of Life)
capable of self-replication and universal computation.
His many other original contributions include
the ultimate extension of the ordered number line:
surreal numbers (1973).

Donald Ervin Knuth established the
rigorous analysis of algorithms as a key aspect of computer science.
Complexity
theory studies the best possible
asymptotic
performance of all procedures that can solve a given problem
(running time and/or memory-space used, as functions of input data size).

Dame Jocelyn Bell Burnell discovered the first pulsar
(neutron star) in July 1967 and the
next three shortly thereafter.
She was then a Ph.D. student supervised by Antony Hewish
(who would be awarded a Nobel prize in physics,
in 1974,
for their subsequent joint work).

He was awarded a Fields Medal (1990) for his mathematical contributions to
a physical theory
(String Theory) which captured the hearts of generations
of physicists without any empirical support.
In 1995, Witten unified the 5 or 6 flavors of that theory under a single umbrella:
M-Theory.