210 Giants of Science
The Entire History of Science at a Glance
Thales of Miletus, engineer (c. 624546 BC)
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].
Electricity

Earliest Mathematics

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Pythagoras of Samos (c. 569475 BC)
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...
Tetractys

Constant of Pythagoras

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Heraclitus of Ephesus (c. 535475 BC)
No man ever steps into the same river twice.
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 Machlike principle of Relativity.
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Parmenides of Elea (c. 515450 BC)
Existence is timeless; change is impossible.
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 (469399 BC).
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Empedocles of Acragas (c. 492432 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.
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Zeno of Elea (c. 490425 BC)
In the most famous of his provocative paradoxes, Zeno asks how
swiftfooted
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.
Tortoise coordinates (GR)

Zeno's arrow

Quantum Zeno effect

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Democritus of Abdera (c. 460370 BC)
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".
PseudoDemocritus alchemical corpus
(still?)

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Hippocrates of Cos, physician (c. 450377 BC)
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.
The 4 Humors of Hippocrates

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Archytas of Tarentum (428347 BC)
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).
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Plato (427347 BC)
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.
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Eudoxus of Cnidus (408355 BC)
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.
Spheres
of Eudoxus by J.L.E Dreyer

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Aristotle of Stagira, logician (384322 BC)
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).
Classical elements

Plenism

Aristotelian mechanics

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Euclid of Alexandria (c. 325265 BC)
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.
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Aristarchus of Samos (c. 310230 BC)
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).
Head of Aristotle's Peripatic School (c. 287 BC)

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Ctesibius of Alexandria (c. 310222 BC)
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 compressedair catapult,
a water organ and the
regulated waterclock
(fed by an overflowing vessel).
Ktesibios

Antikythera Mechanism (c. 70 BC)

Ktesibios Award (MCA, 2000)

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Archimedes of Syracuse (c. 287212 BC)
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.
Lever

Spiral

Parabola

Sand Reckoner

"The Method"

Historical Tidbits

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Eratosthenes of Cyrene (276194 BC)
Eratosthenes headed the
Library of Alexandria
after Apollonius of Rhodes.
In number theory, he is remembered for the
Sieve of Eratosthenes.
He also came up with the first accurate measurement of the
circumference of the Earth.
Armillary sphere

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Apollonius of Perga (262190 BC)
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.
NO PORTRAIT

Circles of Apollonius

Polarity

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Hipparchus of Nicaea (c. 190126 BC)
Hipparch founded trigonometry
(table of chords,
spherical coordinates)
and discovered the
precession
of the equinoxes (130 BC).
The nova of 134 BC inspired him
to compile a catalog of 1080 stars.
His lunar and solar models were accurate enough to predict eclipses.
Magnitude of Stars

Astrolabe

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Hero of Alexandria, physicist (c. AD 1075)
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 lightrays came from the eyes.
Heron's formula (Metrica, c. AD 50)

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Pliny the Elder, encyclopedist (AD 2379)
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.
Historia
Naturalis (Bill Thayer) =
The
Natural History (Bostock & Riley)

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Dioscorides, pharmacologist (c. AD 4090)
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.
De Materia Medica

Greek Medicine

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Menelaus of Alexandria (c. AD 70135)
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.
Sphaerica (c. AD 100)

Menelaus' theorem

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Ptolemy of Alexandria (c. AD 87165)
Claudius Ptolemaeus was a Roman citizen
who wrote in Greek. His first name is unknown
(it's been guessed to be Tiberius).
His Almagest
(c. AD 150) dominated astronomy for many centuries.
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Galen of Pergamos, physician (AD 129217)
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.
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Mary the Jewess, alchemist (3^{rd} century AD)
Earliest female experimentalist on record (signing
Miriam the prophetess, sister of Moses)
she invented the tribikos still
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).
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Maria Prophetissa

Opus Mulierum

Axiom of Maria

Chrysopoeia
(1964) by
Leonora Carrington
Diophantus of Alexandria (c. AD 200284)
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.
Problems from Arithmetica
by Tinka Davis (MS Thesis, 2010)

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Pappus of Alexandria (c. AD 290350)
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 (15771643) in
Centrobaryca (1635) pertaining to
the surface area and the volume of a solid of revolution.
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Hypatia,
neoplatonist
martyr (c. AD 360415)
Daughter of the mathematician
Theon (c. 335405)
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.
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Aryabhata the Elder (AD 476550)
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...
Aryabhata satellite (ISRO, 1975)

Universität Klagenfurt

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Brahmagupta Bhillamalacarya (AD 598668)
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.
Brahmagupta's formula (c. 620)

Identity

Brahmagupta's theorem

Rational quadrilaterals

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Geber, experimental chemist (c. AD 721815)
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.
khemeia

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Jabir

Chemical
Heritage Foundation

al Shindagah
AlKhwarizmi, Algorismus
(c. AD 783, fl.847)
Aljabr (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.
Decimal numeration

Quadratic formula

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Alhazen, "First Scientist" (9651039)
Abu Ali Muhammed ibn alHasn ibn alHaytham alBasri
was hired by AlHakim
and had to feign madness to avoid impossible engineering duties,
until the "Mad Caliph" died (1021).
Early proponent of the scientific method,
Alhazen pioneered optics and anticipated Newton's first law.
De Aspectibus
(1015)

Pinhole camera

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Abu Rayhan alBiruni, Alberonius (9731048)
Celebrated polymath who was first exposed to mathematics by associating with
Abu Nasr Mansur (9701036)
of sine law fame.
AlBiruni pioneered scientific methods in astronomy and geology.
First mathematician to point out the limited validity
of Brahmagupta's simplified formula.
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Omar alKhayyám (10481131)
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.
Binomial theorem

KhayyamSaccheri quadrilateral

AlMarja

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Bháscara Áchárya (11141185)
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.
Siddhanta Shiromani (1150)

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Robert Grosseteste (11681253)
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.
De Luce (1235)

Dawn of the Scientific Method

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Leonardo Pisano Fibonacci (11701250)
He ended a mathematical lull of eight centuries in the West.
As a teenager in Algeria, Fibonacci learned the HinduArabic
decimal system
that he would advocate in Europe.
In Liber Abaci (1202) he discussed many
computational puzzles,
including one
about the Fibonacci sequence...
The Fibonacci Series

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Roger Bacon, Franciscan (12141292)
Nicknamed Doctor Mirabilis. He went to the University of Paris to take a degree
(1241) and he started lecturing on Aristotle there (12341247) before returning
to Oxford. Influenced by Grosseteste, Roger Bacon became the
most active early proponent of the scientific method in Europe.
The First Scientist: A Life of Roger Bacon

Blackpowder (1249)

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William of Ockham,
friar (c.12881348)
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).
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Levi ben Gershon, Gersonides
(12881344)
Noted talmudist and philosopher
known to the French as Léon de Bagnols
(Magister Leo Hebraeus).
Forefather of Group theory,
he studied permutations for their own sake (1321).
He published the modern proof of the law of sines in 1342.
Jewish Encyclopedia

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Jean Buridan, secular teacher (c.12971358)
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.
Buridanica

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Nicole Oresme, bishop (13231382)
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...
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Madhava of Sangamagrama (13501425)
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.
The Story of Mathematics

Kerala school

Madhava series (c.1400)

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Regiomontanus (14361476)
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.
De triangulis omnimodis (1464)

Angle maximization problem

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Leonardo da Vinci (14521519)
A stellar Renaissance painter,
he left 13000 pages of illustrated notes on science and engineering
(in mirrorimage cursive). He was taught mathematics by
Luca Pacioli (14471517)
as he lived with him in Milan, illustrating
"De divina proportione" (14961498)
with skeletonic polyhedra.
Anchiano (birthplace)

Mona Lisa

Codex Leicester

Birth of linear perspective (1413)

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Nicolaus Copernicus (14731543)
Mikolaj Kopernik attended
Krakow,
Bologna,
Padua and
Ferrara.
Thanks to his uncle,
he became a canon at Frauenberg
(1497) where he would have an
observatory.
Around 1514, he gave
an heliocentric
explanation to
planetary
retrograde motion.
De
revolutionibus (1543)

Copernican revolution

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Copernicium (2010)
Paracelsus, physician (14931541)
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.
The dose makes the poison

Alphabet of the Magi

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Girolamo Cardano (15011576)
First scholar to use negative numbers routinely.
In 1545, he revealed the solution of cubic equations obtained by
del Ferro (14651526)
in 1516 and rediscovered (15350213) by
Tartaglia
(15001557).
It had been extended to quartics, in 1540, by his own assistant
Lodovico Ferrari
(15221565).
Founder of
probability theory

Cardan joint

Ars Magna (1545)

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Ambroise Paré, surgeon (15101590)
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).
Dubious portraits

Biography (French)

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Andries Wijtinck van Wesele (15141564)
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.
De humani corporis fabrica (1543)

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François Viète (15401603)
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.
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Tycho Brahe, astronomer (15461601)
Tyge Ottesen Brahe was from the high Danish nobility.
His Uraniborg observatory,
on Hven island,
cost 1% of the state budget but allowed precise (nakedeye)
observations of planetary positions which made possible the work of
Kepler.
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Simon Stevin, Stevinus (15481620)
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).
De Thiende (1585)

Epitaph of Stevinus (1586)

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John Napier of Merchiston (15501617)
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 (15611630).
History of Logarithms

Discovery of Kepler's third law (1618)

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Galileo Galilei (15641642)
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.
The Gaoileo Project (Rice University)

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Johannes Kepler (15711630)
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.
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William Harvey, physician (15781657)
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 (15141564)
had started encouraging students to observe
rather than conform to the precepts of Galen.
Encyclopedia of Science
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Gérard Desargues (15911661)
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.
Desargues graph

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Albert Girard, engineer (15951632)
Frenchborn mathematician who took refuge in Holland (he was a Calvinist).
Generalizing Viète's formulas
to all polynomials, Girard introduced the elementary
symmetric functions of the roots of a normalized
polynomial as the coefficients of that polynomial.
Spherical excess

Fibonacci recursion

Triangle of extraction

Early reciprocity (1632)

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René Descartes (15961650)
Descartes attended the famous Jesuit college of
La Flèche
from 1607 to 1615. He met his scientific mentor
Isaac Beeckman (15881637)
in 1618. He introduced cartesian geometry in one of the three appendices
to Discours
sur la méthode (1637).
La Géométrie

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Bonaventura Cavalieri (15981647)
In Pisa, Cavalieri was mentored by
Benedetto Castelli (15781643)
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...
Cavalieri quadrature

Galileo

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Pierre de Fermat (16011665)
Fermat attended
Toulouse and
Bordeaux,
got a law degree from
Orléans
and purchased an office at the
parlement of Toulouse
in 1631. He pursued investigations in
mathematics
and physics in his spare time
(his judicial work suffered).
Fermat's Little Theorem

Fermat's Last Theorem

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John Wallis (16161703)
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.
Infinity symbol

Wallis' integral

Wallis product, 1655
(Quantum proof, 2015)

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Blaise Pascal (16231662)
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.
Pascal's hexagram theorem (1639)

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Christiaan
Huygens (16291695)
He improved lensmaking (1654)
discovered Titan (1655)
described Saturn's rings (1656) invented the
pendulum clock (1656) and
achromatic eyepieces (1662).
He formulated the centrifugal law (deducing
the inversesquare law of gravity) &
conservation of momentum.
Wave theory of light (1678).
Spacecraft (2005)

Impacts

FRS 1663

Académie des sciences 1666

Tutor of Leibniz 1672

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Takakazu Seki [Kowa] (16421708)
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 (16641739).
Origins

Wasan

Ellipse circumference (approximation)

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Sir Isaac Newton (16431727)
Lucasian professor of
mathematics in 1669. FRS in 1672. Publishes
Principia
in 1687. Retires from research in 1693. Warden (1696) then Master (1699) of the
Royal Mint.
President of the
Royal Society from 1703. Knighted in 1705.
Proponent of the corpuscular theory of light.
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Gottfried Wilhelm Leibniz (16461716)
A major philosopher and a polymath,
Leibniz invented differential calculus
independently
of Newton. He introduced a consistent notation for
integrals and
infinitesimals (1675).
Unlike d'Alembert or Cauchy,
Leibniz didn't think of derivatives
as limits (cf. Robinson).
Against Atomism

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Pierre Varignon, jesuit (16541722)
He reformulated statics (1687) and introduced the notion of torque.
With Guillaume de l'Hôpital,
he pioneered calculus in France.
First holder of the chair of mathematics at the
Collège des
quatrenations (1688).
Inventor of the manometer (1705).
Varignon rejected divergent series.
Torque

Varignon parallelogram

Hyperbolic spiral

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Jacob [Jacques] Bernoulli (16551705)
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.
Law of large numbers

Bernoulli numbers

Ars Conjectandi

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Johann Bernoulli (16671748)
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 workforhire.
L'Hôpital's rule

Catenary

Bracchistochrone (1696)

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Brook Taylor (16851731)
He invented the calculus of finite differences and
integration by parts.
In 1772,
Lagrange would place
Taylor's theorem
at the root of differential calculus.
In discussing the stretched string (1712) Taylor himself stressed the need for
functions without a Taylor expansion!
Taylor expansion (1712)

Linear perspective (1715,1719)

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Maupertuis (16981759)
PierreLouis 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.
Système de la Nature
(1751, anticipating Mendel's genetics)

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Daniel Bernoulli (17001782)
The feuding Bernoulli family
produced five leading Swiss mathematicians, born in 1655, 1667, 1695, 1700 and 1710.
Pioneer of fluid dynamics,
Daniel formulated
Bernoulli's Law
(the cornerstone of aircraft wing design).
His solution of the St Petersburg paradox
helped define utilities (1731).
Bessel functions

Atomic origin of pressure
(1738)

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Benjamin Franklin (17061790)
At the same time as
Watson
(1746) Franklin
formulated the law of conservation
of charge by positing opposite signs for
resinous () and vitreous (+)
electricity.
One of Franklin's many famous quotes

Electric Kite (1752)

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Emilie du Châtelet (17061749)
At 19, GabrielleEmilie de Breteuil married the Marquis
FlorentClaude 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.
Breteuil ring
in the French West Indies 
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Leonhard Euler (17071783)
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
(17551826) who joined in 1773.
The Euler Archive

Tercentenary

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Laura Bassi, physicist (17111778)
Gabriele Manfredi
(16811761)
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.
Stanford (20120104)

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Alexis Clairaut (17131765)
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.
Home

Clairaut's equation (1734)

Théorie de la figure de la Terre
(1743)

Parisian statue

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JeanleRond d'Alembert (17171783)
Editor of the momentous
Encyclopédie.
Born illegitimately to Louis
Camus
des Touches "Canon" (16681726)
and Claudine de Tencin.
He founded analytical mechanics on a principle of
virtual work and solved the wave equation.
He mentored Laplace.
The d'Alembertian
is a 4D operator.
Remarkable
Mathematicians

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MariaGaëtana Agnesi (17181799)
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).
Witch of Agnesi (curve)

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Etienne Bézout,
algebraist (17301783)
His 6volume mathematical textbook (17701782) 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 (xa).
Ph.D. 1756

Lemma

Domain

Bézout's theorem

Bézoutian

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Henry Cavendish, FRS (17311810)
Retrodiagnosed with
Asperger's syndrome, absentminded
and pathologically shy,
he could not talk to women at all.
In 1766, Cavendish discovered what
Lavoisier would call hydrogen.
In 1798,
he measured Newton's Universal constant of gravity
(G) to an accuracy of 1%.
Torsion balance of
John Michell (17241793)

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Joseph Louis Lagrange (17361813)
In 1760, Lagrange tackled the calculus of variations
(named by Euler in 1766). He
applied it to analytical mechanics and
invented Lagrange multipliers (1788).
He gave accurate
secular variations of solar orbits (1782).
Lagrange was the first professor of analysis at
Polytechnique (17941799).
Remarkable
Mathematicians (pdf)

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AntoineLaurent de Lavoisier (17431794)
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.
Chemical
Heritage Foundation 
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Alessandro Volta (17451827)
Correctly interpreting the 1791 observation by
Luigi Galvani (17371798) 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.
(Misleading)
Google Doodle

Electrochemistry

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Gaspard Monge (17461818)
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).
Brouette de Monge (optimal transport)

Lines of curvature 1776

Disphenoids 1809

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Pierre Simon Laplace (17491827)
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).
Taupe Laplace (Caen)

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Edward Jenner, immunologist (17491823)
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.
Vaccination (1796)

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AdrienMarie Legendre (17521833)
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).
Legendre symbols
&
polynomials

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Joseph Fourier (17681830)
In January 1795, JeanBaptiste Joseph Fourier was the star trainee in the new
Ecole normale de l'an III (the forerunner of
ENS)
as he was simultaneously teaching at Polytechnique.
He is the founder of
Harmonic Analysis
(cf. Fourier transform).
PhD / ENS

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Thomas Young, polymath (17731829)
Notorious for his
twoslit experiment
demonstrating the wavelike nature of light (1802)
and for Young's modulus of elasticity (1807).
Young's rule gives the posology for an nyear old child as
n/(n+12) the adult dose.
Young paved the way for the decoding of hieroglyphics by
Champollion.
MD 1795? (Edinburgh)

Ph.D. 1796 (Göttingen)

Young's equation

YoungDupré equation (1805)

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AndreMarie Ampere (17751836)
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.
Ampere's law (1825 &
1861)

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Sophie Germain (17761831)
At 13, the story of the death of Archimedes
inspired her to become a mathematician.
She was 18 when Polytechnique opened
(it was maleonly 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).
Father

Chladni patterns

Sophie Germain primes

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Carl Friedrich Gauss (17771855)
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.
Quadratic reciprocity

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Siméon Poisson (17811840; X1798)
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).
Poisson distribution

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François Arago (17861853; X1803)
He taught analysis and geometry at Polytechnique from 1810 to 1830,
at the peak of his creativity (electromagnet, 1820).
A popular leftwing deputy elected in 1830, Arago became Minister of Marine and War in 1948 and
was instrumental in abolishing slavery in the French Colonies (1848).
X

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Joseph von Fraunhofer (17871826)
In 1814, his observation of the Sun's darkline 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).
Spectroscope (1814)

Diffraction grating (1821)

Fraunhofer diffraction
(equation)

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Augustin Fresnel (17881827; X1804)
Trained in
Caen (18011804) 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.
Equations (1821)

2015 Celebrations

Born in Broglie, raised in Mathieu

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JeanVictor Poncelet (17881867; X1807)
POW in Russia for 15 months (18121814) 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.
Cyclic points

Euler's circle

Porism

Unit (980.665 W)

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Augustin Cauchy (17891857; X1805)
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).
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August Möbius (17901868)
Like his mentor Karl Mollweide,
he was both an astronomer and a mathematician.
He invented homogeneous coordinates (1827)
and gave his name to many concepts:
Möbius plane,
Möbius group,
Möbius function,
inversion formula, etc.
He established angles as signed quantities.
Ph.D. 1815

Möbius strip
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Michael Faraday (17911867)
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.
Diamagnetism (1845)

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Charles Babbage (17911871)
He was Lucasian Professor
(18281839) 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).
Babbage Pages

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Michel Chasles (17931880; X1812)
Professeur of geodesy at Polytechnique from 1841 to 1851,
he inaugurated the Sorbonne
chair of projective geometry,
then called higher geometry (18461867).
His reputation as a science historian was all but ruined when he
bought forged manuscripts (18611869)
from Denis VrainLucas.
Ph.D. 1814

Chasles' theorem

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Jakob Steiner, Swiss geometer (17961863)
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.
Ph.D. 1821

Power (1826)

Constructions (1833)

Systems (1853)

Points

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Niels Henrik Abel (18021829)
Niels Abel
produced many brilliant results during a short life spent in poverty:
Nonsolvability 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.
Abel Prize

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Carl Gustav Jacob Jacobi (18041851)
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.
Ph.D. 1825

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Peter Gustav Lejeune Dirichlet (18051859)
Johann Peter Gustav LejeuneDirichlet.
signed Gustav Lejeune Dirichlet, (no hyphen)
published as P.G.L. Dirichlet
and was quoted as LejeuneDirichlet. He contributed to
number theory, mechanics and
analysis.
He was the first to consider unrestricted functions.
h.c. 1827

Theorems

Dbranes

DE

Life and Work (pdf)

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Sir William Rowan Hamilton (18051865)
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.
DIT 2005

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Charles Robert Darwin (18091882)
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.
The Origin of Species
by Means of Natural Selection (1859)

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Joseph Liouville (18091882; X1825)
Many of Liouville's 400+ papers include key contributions, like his
conservation
of Hamiltonian phasemeasure. In 1836, he founded the
Journal de mathématiques pures et appliquées and promoted
the work of others, including the late Evariste Galois.
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Hermann Grassmann (18091877)
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).
Ph.D. 1840

Grassmann's formula

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Ernst Eduard Kummer (18101893)
Eduard Kummer was Kronecker's inspirational highschool
teacher. He had 55 doctoral students, including Frobenius and
Hermann Schwarz (his soninlaw).
He proved FLT
for all regular primes
and invented the ideal numbers which prompted
Dedekind to build the theory of ideals.
Kummer transformations

GaussKummer series

Ph.D. 1831

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Evariste Galois (18111832)
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.
Numericana

Archives

Galois theory

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Ludwig Schläfli, Swiss geometer (18141895)
He introduced the notion of higherdimensional vectors
(between 1850 and 1852, full treatise published in 1901).
He pioneered multidimensional Riemannian manifolds by considering the
3Dhypersurface of an 4Dhypersphere.
Schläfli also classified
all regular polytopes.
Ph.D. 1860

Schläfli symbols

Schläfli integral (1871)

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James J. Sylvester (18141897)
Sylvester made fundamental contributions to matrix theory,
invariants, number theory,
partitions and
combinatorics.
He inaugurated the chair of mathematics at
Johns Hopkins (18771883)
and founded the
American Journal
of Mathematics (1878). Then, he became
Savilian Professor.
Ph.D. 1841
(Dublin)

Discriminant (1851)

"Law of Inertia" (1852)

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Karl Weierstrass, analyst (18151897)
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.
Hon. Dr. 1854

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George Boole, logician (18151864)
Boole shares credit with
Augustus De Morgan (18061871)
(author of Formal Logic, 1847) for
Boolean logic,
now a fundamental ingredient in digital electronics.
He also published about differential equations.
His wife Mary
(niece of G. Everest)
and daughter Alicia
were mathematicians too.
Mathematical
Analysis of Logic (1847)

Laws of Thought (1854)

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Ada Byron, Lady Lovelace (18151852)
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 ]
Women in Computer Science

Yale CS

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George Gabriel Stokes (18191903)
A former Senior Wrangler,
Sir George Stokes was
Lucasian Professor of Mathematics
at Cambridge for 53 years. He was made a baronet in 1889.
He pioneered advances in fluid dynamics, wave propagation, diffraction,
fluorescence,
differential forms and
divergent series.
[Stokes line]
B.A. 1841

Stokes' theorem

NavierStokes

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Pafnuty Lvovich Chebyshev (18211894)
Homeschooled Russian aristocrat. His mathematics tutor was the textbook author
Platon Nikolaevich Pogorelski (18001852).
Chebyshev contributed to number theory, algebra, analysis, mechanics, etc.
In 1850, he derived
Bertrand's postulate
from the
totient function's asymptotics.
Ph.D. 1849

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Orthogonal polynomials.

Economization
Arthur Cayley, mathematician (18211895)
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 CayleyHamilton theorem:
A matrix is a zero of its characteristic polynomial.
Dr Sc. 1875

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Group Th.

CayleyDickson

Length of a flat ellipse
Hermann von Helmholtz (18211894)
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).
Ph.D. 1842

Energy conservation (1847)

Resonance (1850)

Coil

Equation

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Charles Hermite
(18221901; X1842)
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.
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Louis Pasteur,
microbiologist (18221895)
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.
Pasteur Institute

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Leopold Kronecker, algebraist (18231891)
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.
Ph.D. 1845

Legendre symbols

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Lord Kelvin (18241907)
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).
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Bernhard Riemann, mathematician (18261866)
In 1851, his thesis introduced Riemann surfaces.
Riemann's habilitation lecture on the foundations
of geometry (1854) stunned even Gauss.
Probing the distribution of primes
with his zeta function,
he stated the Riemann Hypothesis in 1859.
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Joseph Lister, surgeon (18271912)
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 postoperative infections and made surgery safer.
Baronet in 1883, he became a Baron in 1897.
Listerine (1879)

Lister Institute (1891)

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James Clerk Maxwell (18311879)
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 MaxwellBoltzmann
kinetic theory of gases.
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Paar (Zagreb)

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Richard Dedekind, mathematician (18311916)
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.
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Dmitri Mendeleev, chemist (18341907)
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).
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Eugenio Beltrami, Italian geometer (18351900)
Bringing to a great conclusion the works of
Gauss,
Bolyai,
Lobachevski and
Riemann
on nonEuclidean 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).
Ph.D. 1856

BeltramiKlein disk

BeltramiPoincaré disk

Houël translation

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M. E. Camille Jordan (18381922; X1855)
A universal mathematician and one of the greatest teachers of the 19th 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.
Ph.D. 1860

Jordan curve theorem

Jordan normal form

JordanHölder theorem

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Ernst Mach, physicist (18381916)
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.
Ph.D. 1860

Mach's principle

Stanford

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Ernst Abbe, optician (18401905)
Founder of modern optics.
His industrial commitments to the instrumentmaker
Carl Zeiss (1816+1888) and
the glassmaker Otto Schott (18511935)
prevented Abbe from accepting a professorship at
Berlin
(offered by Helmholtz).
Ph.D. 1861, 1863

Numerical aperture

Abbe sine condition

Abbe number

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Gaston Darboux, geometer (18421917)
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.
Ph.D. 1866

Darboux sums (1875)

DarbouxRibeaucour trihedron

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John Strutt, Lord Rayleigh (18421919)
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.
Rayleigh's criterion

Ph.D. 1868

Nobel 1904

Encyclopedia

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Sophus Lie, mathematician (18421899)
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
20th century mathematics, with applications to
quantum mechanics.
Ph.D. 1872

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Ludwig Boltzmann, physicist (18441906)
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.
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Georg Cantor, mathematician (18451918)
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.
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William Clifford,
geometer (18451879)
Like Cavendish, Sylvester,
Kelvin and Maxwell before him,
Clifford was Second Wrangler at
Cambridge (in 1867).
"On the SpaceTheory of Matter" (18641876)
Clifford algebras.
CliffordKlein forms.
Ph.D. 1868

Geometric algebra (1876)

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Thomas
Edison, inventor (18471931)
The most successfull
inventor ever.
His 1093 US patents cover the phonograph, lightbulb, 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.
"Hello" (1877)

Edison Birthplace Museum

Edison's homepage by Gerald Beals

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Wilhelm Killing, (18471923)
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).
Ph.D. 1872

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C. Felix
Klein, mathematician (18491925)
Born on 1849425 (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.
Ph.D. 1868

Erlangen program (1872)

Klein bottle

Klein group

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F. Georg
Frobenius (18491917)
In 1892, Weierstrass made him succeed Kronecker
in Berlin,
upholding traditions that would lose out to what flourished at
Göttingen
under Klein.
He contributed to pure mathematics in group theory
(character theory), differential equations, etc.
He proved the CayleyHamilton theorem in 1878.
Frobenius method

Frobenius map

FrobeniusStickelberger formulae

Ph.D. 1870

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Sofia Kovalevskaya (18501891)
Sofia Vasilyevna Kovalevskaya was born Sonya KorvinKrukovskaya.
Weierstrass tutored her privately (18701874) and helped her
become the first female professor at a European university (Stockholm, 1889)
since the days of Laura Bassi (1776) or
MariaGaëtana Agnesi.
Ph.D. 1874

CauchyKovalevskaya theorem (1874)

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Oliver
Heaviside (18501925)
His innovations, which made highermathematics easier to use, include
operational calculus and
vector calculus (which reduced to 4 the number of
Maxwell's equations).
In 1902, he predicted the
KennellyHeaviside layer of the ionosphere,
whose detection (1923) got Appleton a Nobel prize, in 1947.
Electromagnetic terms

LorentzHeaviside units

Heaviside step function

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A. Henri Becquerel (18521908; X1872)
His grandfather,
father
and son had the same career path as himself:
Polytechnician (*), physics chair at the
Muséum national d'histoire naturelle
and member of the
Académie des sciences.
For his discovery of natural
radioactivity (18960301) he shared a Nobel Prize
with Pierre and Marie Curie.
(* Edmond, his father, didn't
attend Polytechnique)

Nobel 1903

SI unit of activity (Bq)

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Gregorio RicciCurbastro (18531925)
In 1884, he started the investigations of quadratic differential forms which led him
to invent tensor calculus
(18841894). The text he published about that with
Tullio LeviCivita
in 1900 would enable Einstein to formulate
General Relativity in 1915.
Ph.D. 1873

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Hendrik A. Lorentz, physicist (18531928)
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).
Nobel 1902

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J. Henri Poincaré (18541912;
X1873)
Doctoral student of Hermite
(1879)
and last universal genius. Quintessential
absentminded professor (cf. Savant Cosinus
comic strip).
Poincaré conceived Special Relativity
before Einstein did. His mathematical legacy includes
chaos theory and contributions to topology.
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Nikola Tesla (18561943)
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.
Visions

Master of Lightning

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Heinrich Hertz (18571894)
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.
Founder of
contact mechanics (1882)

Uncle of Gustav Hertz (Nobel 1925)

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Max Planck, physicist (18581947)
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.
Nobel 1918

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David Hilbert, mathematician (18621943)
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.
Hilbert's List

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Hermann Minkowski (18641909)
Pioneering convex geometry, he proved
an early version of the separation theorem (of HahnBanach)
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 relativistic scalar product (1908)
are named after him.
BrunnMinkowski

HasseMinkowski

Functional

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Jacques Hadamard, analyst (18651963)
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.
Hadamard matrices (1893)

Ph.D. 1892

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Marie Curie, physical chemist (18671934)
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).
Nobel 1903
(Physics)

Nobel 1911
(Chemistry)

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Henrietta S. Leavitt, astronomer (18681921)
In 1908, Henrietta Swan Leavitt
published the periodluminosity 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).
1777
Variables in the Magellanic Clouds (1908)

Calibration by Hertzsprung (1913)

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Felix Hausdorff, topologist (18681942)
In a Hausdorff space (1914) two distinct points
are always disconnected.
In 1919, he introduced fractional dimensions
and defined ddimensional measures.
Hausdorff published literary work as Paul Mongré.
Unable to escape the Nazis, he committed suicide with his wife and sisterinlaw.
Hausdorff gap (1909)

Hausdorff distance

Hausdorff dimension

Hausdorff measure

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Elie Cartan, mathematician (18691951)
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
(19042008).
Ph.D. 1894

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Emile Borel, mathematician (18711956)
A SainteBarbe 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 Topology and
founded Measure theory.
Elected to the Académie des Sciences in 1921.
Ph.D. 1893

HeineBorel criterion

Borelian tribe (1898)

Divergent series (1899)

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Ernest Rutherford (18711937)
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 GeigerMarsden
experiment (1909) and inferred the planetary model of the atom (1911).
Nobel 1908 
Nuclear Physics

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Constantin Carathéodory (18731950)
Greek mathematician with a doctorate from
Göttingen
(under Minkowski).
He made contributions to the
calculus of variations and founded
axiomatic thermodynamics.
In measure theory,
Carathéodory's criterion
characterizes measurability.
He corresponded with
Einstein (19161930).
Ph.D. 1904

Outer measure

Axiomatic thermodynamics (1909)

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René Baire, French analyst (18741932)
Entered
ENS at
17, by derogation (1892). Agrégé at 20.
The Baire space is the set of all infinite sequences of
natural integers,
endowed with the Tychonoff topology.
It's homeomorphic to the
subspace of the
interval [0,1]
consisting of irrational numbers
(cf. continued fractions).
Teaching career (French)

Ph.D. 1899

Baire property

Baire spaces
vs.
Baire space

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Henri Lebesgue, French analyst (18751941)
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.
Ph.D. 1902

Lebesgue spaces

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G.H. Hardy, pure mathematician (18771947)
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.
M.A. 1903

A Mathematician's Apology (1940)

Divergent Series (1949, posthumous)

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Lise Meitner, physical chemist (18781968)
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).
Otto
Hahn's Nobel Lecture

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Meitnerium (1997)
Albert Einstein, physicist (18791955)
In 1905, Einstein published on
Brownian motion (existence of atoms) the photoelectric effect (discovery of the photon)
and his own
Special Theory of Relativity,
which he unified with gravity in 1915 by
formulating the General Theory of Relativity.
In 1916, he
discovered what led to
lasers.
Nobel 1921 
EPR 1935

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Emmy Noether, mathematician (18821935)
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).
1918 Paper

Noether's theorem
&
Proof

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John E. Littlewood, analyst (18851977)
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.
Senior Wrangler 1905

FRS 1916

Rouse Ball professor, 1928

Miscellany (1953,1986)

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Niels Bohr, physicist (18851962)
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
linearlyevolving quantum states).
Nobel 1922

Wikipedia

Coat of Arms

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"Peter" Hermann Weyl (18851955)
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 (19231938).
Ph.D. 1908

Weyl algebra

Symmetry (1952)

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Erwin Schrödinger, physicist (18871961)
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.
Nobel 1933
(lecture)

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Srinivasa Ramanujan (18871920)
Ramanujan lacked a formal mathematical education but, in 1913, a few of his early results
managed to startle G.H. Hardy (18771947) and
J.E. Littlewood (18851977) who invited him to
Cambridge in 1914.
Ramanujan has left an unusual legacy of brilliant unconventional results.
Degree 1916

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Stefan Banach (18921945)
Pioneer of functional analysis
(Théorie des opérations linéaires, 1932).
His name was given to the main backdrop (Banach spaces) and the 3 fundamental theorems:
HahnBanach
(linear extension & separation),
BanachSteinhaus
(uniform boundedness),
BanachSchauder (open map).
home

Ph.D. 1920
(Banach spaces)

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Louis de Broglie, physicist (18921987)
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.
Nobel 1929

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Wolfgang [Ernst] Pauli,
physicist (19001958)
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."
Nobel 1945

greatgrandfather

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Enrico Fermi, physicist (19011954)
In 1926, Fermi helped formulate the FermiDirac statistics
obeyed by what we now call fermions.
He identified the neutrino in betadecay.
He discovered slow neutrons and the radioactivity they induce.
On December 2, 1942, Fermi produced the first selfsustaining nuclear
chain reaction.
Fermions (1926)

"Neutrino" (1933)

Nobel 1938

Fermilab (1969)

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Werner Heisenberg, physicist (19011976)
In 1925, Werner Heisenberg replaced Bohr's semiclassical 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.
Nobel 1932

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Alfred Tarski, logician (19021983)
In 1924, he gave a nice
definition of infinite sets.
Also due to him are the
BanachTarski Paradox
and the TarskiGrothendieck set theory.
His axioms for elementary
Euclidean geometry (1959) form a system
(unlike anything covering arithmetic)
where every true statement is provable.
Ph.D. 1924

Tarski's undefinability theorem

Stanford

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Paul Adrien Maurice Dirac (19021984)
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).
Genealogy

Nobel 1933

Wikipedia

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Andrey Nikolaevich Kolmogorov (19031987)
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 longterm stability of the
solar system can almost be established
(KAM theorem).
Ph.D. 1925

algorithmic complexity

Trennung (T_{0} axiom)

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Jancsi "John" von Neumann (19031957)
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.
NBG

The Scientific 100

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Henri Cartan, mathematician
(19042008)
Son of Elie Cartan (18691951).
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.
Ph.D. 1928

Caen (19281929)

Bourbaki (1935)

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Tommy Flowers, engineer
(19051998)
In 1944, Thomas Harold Flowers built the first largescale 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).
Wikipedia

Bletchley Park

The design of Colossus by Thomas H. Flowers
Kurt Gödel, logician
(19061978)
The completeness theorem in his dissertation (1929)
states that a statement true in every
model 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.
NBG

Ph.D. 1929

Grave 
Centenary

IAS

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André Weil, mathematician (19061998)
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.
D.Sc. 1928

Uniform spaces (1937)

Weil conjectures (1949)

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Ettore Majorana, physicist (19061938?)
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 10,000page legacy
of notebooks written between 1927 and 1932.
He organized his own disappearance in March 1938.
French video by Etienne Klein

Via Spanisperna

Majorana fermion

Wikipedia
Donald Coxeter, geometer (19072003)
Harold Scott MacDonald Coxeter was a Britishborn 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.
Senior Wrangler 1928

Ph.D. 1931

Coxeter groups

The man who saved geometry

Wikipedia

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Alan Turing, computer scientist (19121954)
Top codebreaker 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.
AlanTuring.net

Jack Copeland

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Paul Erdős, mathematician (19131996)
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.
Pronounce it right

Erdös number

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Laurent Schwartz
(19152002)
"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.
Ph.D. 1943

Bourbakist

Fields Medal 1950

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Convolutions & Distributions
Claude Elwood Shannon, engineer (19162001)
Known as the father of information theory.
A Mathematical Theory of Communication (1948).
ShannonHartley
channel capacity theorem.
NyquistShannon
sampling theorem.
Ph.D. 1948

Statistical entropy (1948)

Information unit (Sh)

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During WW2,
William T. Tutte (pronounced Tut)
broke the Lorenz cipher.
His algorithms motivated Flowers' Colossus.
He developed Whitney's
(1935) matroids.
The Tutte graph (1946) disproved
Tait's conjecture (1884).
In 1948, Coxeter invited him to Canada, where he remained.
Tutte polynomial

Tutte theorem

Spring theorem (1963)

Homotopy theorem (1958)

Ph.D. 1948

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Richard P. Feynman, physicist (19181988)
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.
Nobel 1965

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1972 Interviews

1979 QED Lectures

1988

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Abraham Robinson (19181974)
Robinson's nonstandard 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.
Ph.D. 1949

Hyperreal numbers

Nonstandard_analysis

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Benoît Mandelbrot, mathematician (19242010)
Nephew of the founding bourbakist
Szolem
Mandelbrojt (18991983). His family emigrated from Poland to France
in 1936 and he was educated at
Polytechnique.
He founded fractal geometry
and discovered the Mandelbrot set.
Ph.D. 1952

Fractals and Roughness

Wikipedia

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Alexander Grothendieck (19282014)
Visionary mathematician.
Student of Schwartz,
advisor of Deligne.
Championing categories,
he advocated Schemes and Motives.
Having given up research in 1972, he
retired in 1988.
In 1991, he chose to live as a
recluse
in Ariège
(09230 Lasserre,
pop. 211;
16 km North of SaintGirons).
Ph.D. 1953

Grothendieck Circle

Cartier

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John Forbes Nash, Jr.
(19282015)
The notion of a
Nash equilibrium
(in his 1950 dissertation about noncooperative games)
makes game theory relevant
to many reallife 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).
Ph.D. 1950

Embedding

Nobel Prize, 1994

Brilliant madness

A Beaufiful Mind

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John S. Bell, Irish physicist
(19281990)
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 yearlong sabbatical from CERN, John published his masterpiece:
"On the EPR paradox" (1964).
Bell's inequality (1964)

The most profound discovery
in Science

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Steven Weinberg, physicist (1933)
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.]
Ph.D. 1957

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Nobel 1979

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Emperor
Has No Clothes Award

FB Fans

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Paul Cohen, logician (19342007)
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...
Ph.D. 1958 (on USets)

CH (1963)

Fields medal (1966)

The Story of Mathematics

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John H. Conway (1937)
In 1970, Conway found the simple rules of a cellular automaton
(the Game of Life)
capable of selfreplication and universal computation.
His many other original contributions include
the ultimate extension of the ordered number line:
surreal numbers (1973).
bibliography

New York Times

The 3 Conway sporadic groups

Wikipedia

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Don Knuth, computer scientist (1938)
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 memoryspace used, as functions of input data size).
home

The Art
of Computer Programming

Ph.D. 1963 (Caltech)

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Susan Jocelyn Bell, astrophysicist (1943)
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).
Little Green Men (LGM 1,2,3,4)

Encyclopedia Britannica

Wikipedia

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Ed Witten, theoretical physicist (1951)
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:
MTheory.
MTheory

WeinbergWitten theorem

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Sharing Science on the Web

Giants of Science

Solvay Conferences

Armorial
Taupe Laplace

Nicolas Bourbaki

Lucien Refleu

Roger Apéry

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