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Antiquity

Middle Ages

Renaissance

Enlightenment

Modern

Mirzakhani

Perelman

Wiles

Conway

Penrose

Nash

Grothendieck

Mandelbrot

Blackwell

Johnson

Lorenz

Erdős

Turing

Gödel

von Neumann

Escher

Shannon

Ramanujan

Noether

Einstein

Hardy

Russell

Hilbert

Peano

Poincaré

Cantor

Lie

Carroll

Dedekind

Riemann

Cayley

Nightingale

Boole

Lovelace

Galois

Jacobi

Hamilton

Abel

Bolyai

Lobachevsky

Babbage

Cauchy

Gauss

Germain

Fourier

Laplace

Monge

Lagrange

Lambert

Euler

Bernoulli

Goldbach

De Moivre

Bernoulli

Leibniz

Newton

Pascal

Wallis

Fermat

Cavalieri

Descartes

Desargues

Mersenne

Kepler

Galileo

Napier

Stevin

Viète

Cardano

Tartaglia

Copernicus

Da Vinci

Pacioli

Regiomontanus

Madhava

Oresme

Zhu Shijie

Qin Jiushao

Fibonacci

Bhaskara

Khayyam

Al-Haytham

Al-Karaji

Al-Khwarizmi

Brahmagupta

Aryabhata

Hypatia

Diophantus

Ptolemy

Nicomachus

Apollonius

Eratosthenes

Archimedes

Pingala

Euclid

Aristotle

Plato

Democritus

Pythagoras

Thales

c. 300 BC: Indian mathematician Pingala writes about zero, binary numbers,
Fibonacci numbers, and Pascal’s triangle.

c. 260 BC: Archimedes proves that π is between 3.1429 and 3.1408.

c. 235 BC: Eratosthenes uses a sieve algorithm to quickly find prime numbers.

c. 200 BC: The “Suàn shù shū” (Book on Numbers and Computation) is one of the
oldest Chinese texts about mathematics.

c. 100 AD: Nicomachus poses the oldest still-unsolved problem in mathematics:
whether there are any odd perfect numbers.

c. 250 AD: The Mayan culture in Central America flourishes, and uses a base-20
numeral system.

c. 830 AD: Al-Khwarizmi publishes “Kitab al-jabr wa al-muqābalah”, the first
book about – and the namesake of – Algebra.

1202: Fibonacci’s Liber Abaci introduces Arabic numerals to Europe, as well
as simple algebra and the Fibonacci numbers.

1482: First printed edition of Euclid’s Elements

1545: Cardano conceives the idea of complex numbers.

1609: Kepler publishes the “Astronomia nova”, where he explains that planets
move on elliptical orbits.

1618: Napier publishes the first references to the number e, in a book on
logarithms.

1637: Fermat claims to have proven Fermat’s Last Theorem.

1654: Pascal and Fermat develop the theory of probability.

1684: Leibniz’ publishes the first paper on the calculus.

1687: Newton publishes the Principia Mathematica, containing the laws of
gravity and motion, as well as his version of calculus.

1736: Euler solves the Königsberg bridges problem by inventing graph theory.

1761: Lambert proves that π is irrational

1799: Gauss proves the fundamental theorem of algebra.

1829: Bolyai, Gauss and Lobachevsky all invent hyperbolic non-Euclidean
geometry.

1832: Galois finds a general condition for solving algebraic equations,
thereby founding Group theory and Galois theory.

1858: August Ferdinand Möbius invents the Möbius strip.

1874: Cantor proves that there are different “sizes” of infinity, and that
the real numbers are uncountable.

1895: Poincaré’s paper “Analysis Situs” starts modern topology.

1905: Einstein explains the photoelectric effect and Brownian motion,
discovers special relativity, and E = mc².

1915: Noether shows that every conservation law in physics corresponds to a
symmetry of the universe.

1931: Gödel’s incompleteness theorem establishes that mathematics will
always be incomplete.

1939: A group of French mathematicians publish their first book under the
pseudonym of Nicolas Bourbaki, on Set theory.

1961: Lorenz discovers chaotic behavior in weather simulations – the
butterfly effect.

1977: Adelman, Rivest and Shamir introduce public-key cryptography using
prime numbers.

1994: Andrew Wiles proves Fermat’s Last Theorem.

2003: Perelman proves the Poincaré conjecture, the only one of the seven
Millennium problems that have been solved to date.

c. 563 BC: Buddha is born in India. His teachings become the foundation of
Buddhism.

c. 551 BC: Confucius is born in China. His teachings become the foundation of
Confucianism.

490 BC: Greece stop the Persian invasion at the battle of Marathon.

432 BC: The Acropolis is built in Athens, during its golden age under the
rule of Pericles.

399 BC: Socrates is sentenced to death, refuses to escape, and drinks a cup of
poison.

327 BC: Alexander the Great invades India, having created an enormous
empire across Asia.

c. 221 BC: Qin Shi Huang unifies China and starts construction of the Great Wall.

146 BC: The Roman army destroys Carthage, ending the third Punic War.

44 BC: Julius Caesar is murdered.

4 BC: Jesus of Nazareth is born in Bethlehem, establishing Christianity.

180 AD: The death of Marcus Aurelius ends the Pax Romana, a 200 year period
of peace across Europe.

476 AD: Fall of the Roman Empire

570 AD: Muhammad, the founder of Islam, is born in Mecca.

c. 641 AD: The Library of Alexandria is destroyed.

800 AD: Charlemagne is crowned as the first Holy Roman Emperor.

c. 870 AD: Norse explorers discover and colonise Iceland.

1066: William the Conqueror wins the battle of Hastings and is crowned King
of England.

1088: The first university is established in Bologna, Italy.

1096: The first Crusade is launched by Pope Urban II.

1206: Genghis Khan defeats his rivals and receives the title “Universal
Ruler of the Mongols”.

1215: King John of England is forced to sign the Magna Carta, restricting
his powers.

1266: Marco Polo arrives at the court of Kublai Khan in Beijing.

c. 1347: The Black Death kills millions of people across Europe.

1439: Johannes Gutenberg invents the printing press.

1453: The Ottoman Turks conquer Constantinople, marking the fall of the
Byzantine empire.

1492: Christopher Columbus arrives in America, starting a new age of
European conquest.

1517: Martin Luther publishes his 95 theses, starting the Protestant
reformation.

1522: Ferdinand Magellan’s expedition circumnavigates Earth.

1543: Polish scientist Nicolaus Copernicus writes that the Earth revolves
around the Sun.

1588: Under Queen Elizabeth I, England defeats the Spanish Armada.

1603: William Shakespeare’s “Hamlet” is performed for the first time.

1633: Galileo Galilei is tried by the Catholic Inquisition for his
scientific writings.

1649: King Charles I is tried and beheaded during the English Civil War.

1756: Wolfgang Amadeus Mozart is born in Austria.

c. 1765: James Watt invents a more efficient steam engine, that will power
the industrial revolution.

1776: America Issues its Declaration of Independence from Great Britain.

1789: Revolutionaries storm the Bastille in Paris, starting the French
Revolution.

1804: Napoleon is crowned emperor of France.

1819: Simón Bolívar defeats Spain at the Battle of Boyacá, leading to the
independence of many South American countries.

1837: Samuel Morse and others develop electrical telegraphs.

1859: Charles Darwin publishes “On the Origin of Species”, introducing
natural selection.

1865: Abraham Lincoln is assassinated, at the end of the American Civil War.

1876: Alexander Bell invents the telephone.

1903: The Wright Brothers construct the first powered, heavier-than-air
aircraft.

1914: Franz Ferdinand of Austria is assassinated in Sarajevo, starting the
first World War.

1929: The Black Tuesday stock market crash starts the great depression.

1939: Adolf Hitler invades Poland, starting World War II.

1953: Watson and Crick discovery the double helix structure of DNA.

1957: The Soviet Union launches Sputnik 1, the first man-made satellite into
space.

1969: Apollo 11 astronauts Neil Armstrong and Buzz Aldrin land and walk on
the moon.

1975: End of the Vietnam War

1989: Tim Berners-Lee invents the World Wide Web.

**Maryam Mirzakhani** (مریم میرزاخانی, 1977 – 2017) was an Iranian mathematician and professor at Standford University. She is the only woman to have received the *Fields Medal*, the highest award in mathematics.

Maryam worked at the intersection of dynamical systems and geometry. She studied objects like *hyperbolic surfaces* and *complex manifolds*, but also contributed to many other areas of mathematics.

When solving problems, Maryam would draw doodles and diagrams on large sheets of paper, to see the underlying patterns and beauty. Her daughter even described Maryam’s work as “painting”. At the age of 40, Maryam died of breast cancer.

In 2003, the Russian mathematician **Grigori Perelman** (Григо́рий Перельма́нborn, born 1966) proved the *Poincaré Conjecture*, which, until then, was one of the most famous unsolved problems in mathematics.

The complex proof was verified by 2006, but Perelman declined two big awards that came with it: the $1 million Clay Millennium Prize, and the *Fields Medal* which is the highest recognition in mathematics. In fact, he said: *“I’m not interested in money or fame; I don’t want to be on display like an animal in a zoo.”*

Perelman also made contributions to Riemannian geometry and geometric topology, and the Poincaré Conjecture is still the only one of the seven Millennium Prize problems to have been solved.

The British mathematician Sir **Andrew Wiles** (born 1953) is best known for proving Fermat’s Last Theorem, which, until then, was one of the most famous unsolved problems in mathematics.

In 1637, Pierre de Fermat, wrote in the margin of a textbook that he had a wonderful proof that the equation

Wiles had been fascinated by the problem since the age of 10, and spent seven years working on it in solitude. He announced his solution in 1993, although a small gap in his argument took two more years to fix.

He was too old to receive the *Fields medal*, the highest award in mathematics, which has an age limit of 40. Instead, Wiles was awarded a special silver plaque for his work.

**John Horton Conway** (born 1937) is a British mathematician and currently Professor Emeritus at Princeton University.

He is best known for inventing “Conway’s Game of Life”, a *cellular automaton* with fascinating properties. He explored the underlying mathematics of everyday objects like knots and games, as well as contributing to group theory, number theory and many other areas of mathematics.

**Sir Roger Penrose** (born 1931) is a British mathematician and physicist who is known for his groundbreaking work in general relativity and cosmology – often collaborating with other famous scientists like Stephen Hawking and Michael Atiyah. He also discovered *Penrose Tilings*: self-similar, non-periodic tessellations.

**John Forbes Nash** (1928 – 2015) was an American mathematician who worked on game theory, differential geometry and partial differential equations. He showed how mathematics can explain the decision-making in complex, real-life systems – including economics and the military.

In his 30s, Nash was diagnosed with paranoid schizophrenia, but he managed to recover and return to his academic work. He is the only person to receive both the Nobel Prize for economics and the *Abel Prize*, one of the highest awards in mathematics.

The French mathematician **Alexander Grothendieck** (1928 – 2014) was one of the key figures in the development of *algebraic geometry*. He extended the scope of the field to apply to many new problems in mathematics, including, eventually, Fermat’s last theorem. In 1966, he was awarded the Fields medal.

The mathematician **Benoît Mandelbrot** was born in Poland, grew up in France, and eventually moved to the United States. He was one of the pioneers of *fractal geometry*, and particularly interested in how “roughness” and “chaos” appear in the real world (e.g. clouds or coastlines).

While working at IBM, he used early computers to create graphical representations of fractals, and in 1980 he discovered the famous *Mandelbrot set*.

**David Blackwell** (1919 – 2010) was an American statistician and mathematician. He worked on game theory, probability theory, information theory and dynamic programming, and wrote one of the first textbooks on Bayesian statistics. The *Rao-Blackwell Theorem* shows how to improve estimators of certain quantities in statistics.

Blackwell was the first African-American elected to join the American *National Academy of Sciences*, and he was one of the first to receive a PhD in mathematics.

**Katherine Johnson** (born 1918) is an African-American mathematician. While working at NASA, Johnson calculated the orbits taken by American astronauts – including Alan Shepard, the first American in space, the Apollo Moon landing program, and even the Space Shuttle.

Her extraordinary ability to calculate orbital trajectories, launch windows and emergency return paths was widely known. Even after the arrival of computers, astronaut John Glenn asked her to personally re-check the electronic results.

In 2015, Johnson received the Presidential Medal of Freedom.

**Edward Lorenz** (1917 – 2008) was an American mathematician and meteorologist. He pioneered *chaos theory*, discovered *strange attractors*, and coined the term “butterfly effect”.

**Paul Erdős** (1913 – 1996) was one of the most productive mathematicians in history. Born in Hungary, he solved countless problems in graph theory, number theory, combinatorics, analysis, probability, and other parts of mathematics.

During his life, Erdős published around 1,500 papers and collaborated with more than 500 other mathematicians. In fact, he spent most of his life living out of a suitcase, travelling to seminars, and visiting colleagues!

**Alan Turing** (1912 – 1954) was an English mathematician and is often called the “father of computer science”.

During the Second World War, Turing played a critical role in breaking the Enigma code used by the German military, as part of the “Government Code and Cypher School” at Bletchley Park. This helped the Allies win the war, and may have saved millions of lives.

He also invented the *Turing machine*, a mathematical model of a general purpose computer, and the *Turing test*, which can be used to judge the ability of artificial intelligence.

Turing was gay, which was still a crime during his life, and meant that his groundbreaking accomplishments were never fully recognised. He committed suicide at the age of 41.

**Kurt Gödel** (1906 – 1978) was an Austrian mathematician who later immigrated to America, and is considered one of the greatest logicians in history.

At the age of 25, just after finishing his doctorate in Vienna, he published his two *incompleteness theorems*. These state that any (consistent and sufficiently powerful) mathematical system contains certain statements that are true but cannot be proven. In other words, mathematics contains certain problems that are impossible to solve.

This result had a profound impact on the development and philosophy of mathematics. Gödel also found an example of these “impossible theorems”: the *continuum hypothesis*.

**John von Neumann** (1903 – 1957) was a Hungarian-American mathematician, physicist and computer scientist. He made important contributions to pure mathematics, was a pioneer of quantum mechanics, and developed concepts like game theory, cellular automata, self-replicating machines, and linear programming.

During World War II, von Neumann was a key member of the *Manhattan Project*, working on the development of the hydrogen bomb. He later consulted for the Atomic Energy Commission and the US Air Force.

**Maurits Cornelis Escher** (1898 – 1972) was a Dutch artist who created sketches, woodcuts and lithographs of mathematically inspired objects and shapes: including polyhedra, tessellations and impossible shapes. He graphically explored concepts like symmetry, infinity, perspective and non-euclidean geometry.

**Claude Shannon** (1898 – 1972) was an American mathematician and electrical engineer, remembered as the “father of information theory”. He worked on cryptography, including codebreaking for national defence during World War II, but he was also interested in juggling, unicycling and chess. In his spare times, he built machines that could juggle or solve the Rubik’s Cube puzzle.

**Srinivasa Ramanujan** (1887 – 1920) grew up in India, where he received very little formal education in mathematics. Yet, he managed to develop new ideas in complete isolation, while working as a clerk in a small shop.

After a few failed attempts to contact other mathematicians, he wrote a letter to the famous G.H. Hardy. Hardy immediately recognised Ramanujan's genius, and arranged for him to travel to Cambridge in England. Together, they made numerous discoveries in number theory, analysis, and infinite series.

Unfortunately, Ramanujan soon fell ill and was forced to return to India, where he died at the age of 32. During his short life, Ramanujan proved over 3000 theorems and equations, on a wide range of topics. His work created entirely new areas of maths, and his notebooks were studied by other mathematicians for many decades after his death.

**Amalie Emmy Noether** (1882 – 1935) was a German mathematician who made important discoveries in abstract algebra and theoretical physics, including the connection between symmetry and conservation laws. She is often described as the most influential female mathematician.

**Albert Einstein** (1879 – 1955) was a German physicist, and one of the most influential scientists in history. He received the Nobel Prize for physics and TIME magazine called him the *person of the 20th century*.

Einstein triggered the most significant transformation in our view of the universe since Newton. He realised that classical, *Newtonian* physics was no longer enough to explain certain physical phenomenons.

At the age of 26, during his “miracle year”, he published four groundbreaking scientific papers that explained the photoelectric effect and Brownian motion, introduced special relativity, and derived the formula *E*) and mass (*m*) are equivalent.

**G.H. Hardy** (1877 – 1947) was a leading English pure mathematician. Together with *John Littlewood*, he made important discoveries in analysis and number theory, including the distribution of prime numbers.

In 1913, Hardy received a letter from *Srinivasa Ramanujan*, an unknown, self-taught clerk from India. Hardy immediately recognised his genius, and arranged for Ramanujan to travel to Cambridge where he was working. Together, they made important discoveries and authord numerous paper.

Hardy always dislkied applied mathematics and expressed this in his personal account of mathematical thinking, the 1940 book *A Mathematician’s Apology*.

**Bertrand Russell** (1872 – 1970) was a British philosopher, mathematician and author. He is widely considered to be one of the most important logicians of the 20th century.

Russell co-wrote the “Principia Mathematica”, where he attempted to create a formal foundation for mathematics using logic. His work has had a significant impact not just on maths and philosophy, but also on linguistics, artificial intelligence and metaphysics.

Russell was a passionate pacifist and anti-war activist. In 1950, he received the Nobel Prize in Literature, for his work “in which he champions humanitarian ideals and freedom of thought”.

**David Hilbert** (1862 – 1943) was one of the most influential mathematicians of the 20th century. He worked on almost every area of mathematics, and was particularly interested in building a formal, logical foundation for maths.

Hilbert worked in Göttingen (Germany), where he tutored numerous students who later became famous mathematicians. During the International Congress of Mathematicians in 1900, he presented a list of 23 unsolved problems. These set the course for future research – and four of them are still unsolved today!

The Italian mathematician **Giuseppe Peano** (1858 – 1932) published over 200 books and papers about logic and mathematics. He formulated the *Peano axioms*, which became the basis for rigorous algebra and analysis, developed the notation for logic and set theory, constructed continuous, space-filling curves (*Peano curves*), and worked on the method of proof by induction.

Peano also developed a new, international language, *Latino sine flexione*, which was a simplified version of Latin.

The French mathematician **Henri Poincaré** (1854 – 1912) is often described as *the last universalist*, meaning that he worked in every field of mathematics known during his lifetime.

Poincaré is one of the founders of the field of *Topology*, and he came up with the *Poincaré conjecture*. This was one of the famous unsolved problems in mathematics, until it was proven in 2003 by Grigori Perelman

He also found a partial solution for the “three body problem”, and discovered that the motion of three stars or planets in space can be completely unpredictable. This laid the foundation for modern *Chaos theory*.

Poincaré was the first to propose *gravitational waves*, and his work on Lorentz transformations was the basis upon which Albert Einstein built his theory of special relativity.

The German mathematician **Georg Cantor** (1845 – 1918) was the inventor of set theory, and a pioneer in our understanding of infinity. For most of his life, Cantor's discoveries were fiercely opposed by his colleagues. This may have contributed to his depression and nervous breakdowns, and he spent many decades in a mental institution.

Cantor proved that there are *different sizes* of infinity. The set of real numbers, for example, is *uncountable* – meaning that it cannot be paired up with the set of natural numbers.

Only towards the end of his life, Cantor started to receive the recognition he deserved. David Hilbert famously declared that “No one shall expel us from the paradise that Cantor has created”.

The Norwegian mathematician **Marius Sophus Lie** (1842 – 1899) made significant advances in the study of *continuous transformation groups* – now called *Lie groups*. He also worked on differential equations and non-Euclidean geometry.

**Charles Lutwidge Dodgson** (1832 – 1898) is best know under his pen name **Lewis Carroll**, as the author of *Alice’s Adventures in Wonderland* and its sequel *Through the Looking-Glass*.

However, Carroll was also a brilliant mathematician. He always tried to incorporate puzzles and logic into his children’s stories, making them more enjoyable and memorable.

**Richard Dedekind** (1831 – 1916) was a German mathematician and one of the students of Gauss. He developed many concepts in set theory, and invented *Dedekind cuts* as the formal definition of real numbers. He also gave the first definitions of *number fields* and *rings*, two important constructs in abstract algebra.

**Bernhard Riemann** (1826 – 1866) was a German mathematician working in the fields of analysis and number theory. He came up with the first rigorous definition of integration, studied differential geometry which laid the foundation for general relativity, and made groundbreaking discoveries regarding the distribution of prime numbers.

**Arthur Cayley** (1821 – 1895) was a British mathematician and lawyer. He was one of the pioneers of *group theory*, first proposed the modern definition of a “group”, and generalised them to encompass many more applications in mathematics. Cayley also developed matrix algebra, and worked on higher-dimensional geometry.

**Florence Nightingale** (1820 – 1910) was an English nurse and statistician. During the Crimean War, she nursed wounded British soldiers, and later founded the first training school for nurses. As the “The Lady with the Lamp”, she bacame a cultural icon, and new nurses in the US still take the *Nightingale pledge*.

One of her most important contributions to medicine was the use of statistics to evaluate treatments. She created numerous infographics, and was one of the first to use pie charts. Nightingale also worked to improve sanitation and hunger relief in India, helped abolish prostitution laws, and promoted new careers for women.

**George Boole** (1815 – 1864) was an English mathematician. As a child, he taught himself Latin, Greek and mathematics, hoping to escape his lower class life. He created *Boolean algebra*, which uses operators like AND, OR and NOT (rather than addition or multiplication) and can be used when working with sets. This was the foundation for formal mathematical logic, and has many applications in computer science.

**Ada Lovelace** (1815 – 1852) was an English writer and mathematician. Together with Charles Babbage, she worked on the *Analytical Engine* an early, mechanical computer. She also wrote the first algorithm to run on such a machine (to calculate Bernoulli numbers), making her the first computer programmer in history.

Ada described her approach as “poetical science”, and spent much time thinking about the impact of technology on society.

The French mathematician **Évariste Galois** (1811 – 1832) had a short and tragic life, yet he invented two entirely new fields of mathematics: *Group theory* and *Galois theory*.

While still in his teens, Galois proved that there is no general solution for polynomial equations of degree five or higher – simultaneously with Niels Abel.

Unfortunately, other mathematicians who he shared these discoveries with repeatedly misplaced or simply returned his work, and he failed his school and university exams while concentrating on much more complex work.

At the age of 21, Galois was shot in a duel (some say a feud over a woman), and later died of his wounds. In the night before his death, he summarised his mathematical discoveries in a letter to a friend. It would take other mathematicians many years to fully realise the true impact of his work.

**Carl Jacobi** (1804 – 1851) was a German mathematician. He worked on analysis, differential equations and number theory, and was one of the pioneers in the study of *elliptic functions*.

**William Rowan Hamilton** (1805 – 1865) was an Irish mathematician and child prodigy. He invented *quaternions*, the first example of a “non-commutative algebra”, which has important applications in mathematics, physics and computer science.

He first came up with the idea while walking along the Royal Canal in Dublin, and carved the fundamental formula into a stone bridge he passed:

Hamilton also made significant contributions to physics, including optics and Newtonian mechanics.

**Niels Henrik Abel** (1802 – 1829) was an important Norwegian mathematician. Even though he died at the age of 26, he made groundbreaking contributions to a wide range of topics.

At the age of 16, Abel proved the binomial theorem. Three years later, he proved that it is impossible to solve quintic equations – by independently inventing group theory. This had been an open problem for over 350 years! He also worked on elliptic functions and discovered *Abelian* functions.

Abel spent his life in poverty: he had six siblings, his father died when he was 18, he was unable to find a job at a university, and many mathematicians initially dismissed his work. Today, one of the highest awards in mathematics, the *Abel Prize* is named after him.

**János Bolyai** (1802 – 1860) was a Hungarian mathematician, and one of the founders of non-Euclidean geometry – a geometry in which Euclid’s fifth axiom about parallel lines does not hold. This was a significant breakthrough in mathematics. Unfortunately for Bolyai, the mathematicians Gauss and Lobachevsky discovered similar results at the same time, and received most of the credit.

**Nikolai Lobachevsky** (Никола́й Лобаче́вский) was a Russian mathematician, and one of the founders of non-Euclidean geometry. He managed to show that you can build up a consistent type of geometry in which Euclid’s fifth axiom (about parallel lines) does not hold.

**Charles Babbage** (1791 – 1871) was a British mathematician, philosopher and engineer. He is often called the “father of the computer”, having invented the first mechanical computer (the *Difference engine*), and an improved, programmable version (the *Analytical Engine*).

In theory, these machines could automatically perform certain calculations stored on cards or tape. However, due to the high production costs, they were never fully completed during Babbage’s lifetime. In 1991, a functional replica was constructed at the Science Museum in London.

**Augustin-Louis Cauchy** (1789 – 1857) was a French mathematician and physicist. He contributed to a wide range of areas in mathematics, and dozens of theorems and named after him.

Cauchy formalised calculus and analysis, by reformulating and proving results where previous mathematicians were much more careless and imprecise. He founded the field of *complex analysis*, studied permutation groups, and worked on optics, fluid dynamics and elasticity theory.

**Carl Friedrich Gauss** (1777 – 1855) was arguably the greatest mathematician in history. He made groundbreaking discoveries in just about every field of mathematics, from algebra and number theory to statistics, calculus, geometry, geology and astronomy.

According to legend, he corrected a mistake in his father‘s accounting at the age of 3, and found a way to quickly add up all integers from 1 to 100 at the age of 8. He made his first important discoveries while still a teenager, and later tutored many other famous mathematicians as Professor.

**Marie-Sophie Germain** (1776 – 1831) decided that she wanted to be a mathematician at the age of 13, after reading about Archimedes. Unfortunately, as a woman, she was faced with significant opposition. Her parents tried to prevent her from studying when she was young, and she never received a post at a university.

Germain was a pioneer in understanding the mathematics of elastic surfaces, for which she won the grand prize from the Paris Academy of Sciences. She also made considerable progress in solving Fermat’s Last Theorem, and regularly corresponded with Carl Friedrich Gauss.

**Joseph Fourier** (1768 – 1830) was a French mathematician, and a friend and advisor of Napoleon. In addition to his mathematical research, he is also credited with the discovery of the greenhouse effect.

While travelling to Egypt, Fourier became particularly fascinated with *heat*. He studied heat transfer and vibrations, and discovered that any periodic function can be written as an infinite sum of trigonometric functions: a *Fourier series*.

**Pierre-Simon Laplace** (1749 – 1827) was a French mathematician and scientist. He is sometimes called the “Newton of France”, because of his wide range of interests, and the enormous impact of his work.

In a five-volume book, Laplace translated problems in celestial mechanics from *geometry* to *calculus*. This opened up a wide range of new strategies for understanding our universe. He proposed that the solar system developed from a rotating disk of dust.

Laplace also pioneered the field of probability, and showed how probability can help us understand data from the physical world.

**Gaspard Monge** (1746 – 1818) was a French mathematician. He is considered the father of *differential geometry*, having introduced the concept of *lines of curvature* on surfaces in three-dimensional space (e.g. on a sphere). Monge also invented *orthographic projection* and *descriptive geometry*, which allows the representation of three-dimensional objects using two-dimensional drawings.

During the French Revolution, Monge served as Minister of the Marine. He helped reform the French education system and found the École Polytechnique.

**Joseph-Louis Lagrange** (1736 – 1813) was an Italian mathematician who succeeded Leonard Euler as the director of the Academy of Sciences in Berlin.

He worked on analysis and the calculus of variations, invented new methods for solving differential equations, proved theorems in number theory, and laid the foundations of group theory.

Lagrange also wrote about classical and celestial mechanics, and helped establish the metric system in Europe.

**Johann Lambert** (1728 – 1777) was a Swiss mathematician, physicist, astronomer and philosopher. He was the first to prove that *π* is an irrational number, and he introduced hyperbolic trigonometric functions. Lambert also worked on geometry and cartography, created map projections, and foreshadowed the discovery of non-Euclidean spaces.

**Leonhard Euler** (1707 – 1783) was one the greatest mathematicians in history. His work spans all areas of mathematics, and he wrote 80 volumes of research.

Euler was born in Switzerland and studied in Basel, but lived most of his life in Berlin, Prussia, and St. Petersburg, Russia.

Euler invented much of the modern mathematical terminology and notation, and made important discoveries in calculus, analysis, graph theory, physics, astronomy, and many other topics.

**Daniel Bernoulli** (1700 – 1782) was a Swiss mathematician and physicist. He was one of the many famous scientists from the Bernoulli family – including his father Johann, his uncle Jacob, and his brother Nicholas.

Daniel Bernoulli showed that as the speed of a fluid increases, its pressure decreases. Now called *Bernoulli’s principle*, this is the mechanism used by airplane wings and combustion engines. He also made important discoveries in probability and statistics, and first encountered *Bessel functions*.

At the age of 34, he was banned from his father’s house for beating him at an award from the Paris Academy, for which they both submitted an entry.

**Christian Goldbach** (1690 – 1764) was a Prussian mathematician and contemporary of Euler, Leibniz and Bernoulli. He was tutor of Russian Tsar Peter II, and is remembered for his “Goldbach Conjecture“.

**Abraham de Moivre** (1667 – 1754) was a French mathematician who worked in probability and analytic geometry. He is most remembered for *de Moivre’s formula*, which links trigonometry and complex numbers.

De Moivre discovered the formula for the normal distribution in probability, and first conjectured the *central limit theorem*. He also found a non-recursive formula for Fibonacci numbers, linking them to the golden ratio

**Jacob Bernoulli** (1655 – 1705) was a Swiss mathematician, and one of the many important scientists in the Bernoulli family. In fact, he had a deep academic rivalry with several of his brothers and sons.

Jacob made significant advances to the calculus that was invented by Newton and Leibnitz, created the field of *calculus of variations*, discovered the fundamental constant *e*, developed techniques for solving differential equations, and much more.

He published the first substantial work about probability, including permutations, combinations and the law of large numbers, he proved the binomial theorem, and derived many of the properties of Bernoulli numbers.

**Gottfried Wilhelm Leibniz** (1646 – 1716) was a German mathematician and philosopher. Among many other achievements, he was one of the inventors of calculus, and created some of the first mechanical calculators.

Leibniz believed that our universe is the “best possible universe” that God could have created, while allowing us to have a free will. He was a great advocate of *rationalism*, and also made contributions to physics, medicine, linguistics, law, history, and many other subjects.

**Sir Isaac Newton** (1642 – 1726) was an English physicist, mathematician, and astronomer, and one of the most influential scientists of all time. He was a professor at Cambridge University, and president of the Royal Society in London.

In his book *Principia Mathematica*, Newton formulated the laws of motion and gravity, which laid the foundations for classical physics and dominated our view of the universe for the next three centuries.

Among many other things, Newton was one of the inventors of calculus, built the first reflecting telescope, calculated the speed of sound, studied the motion of fluids, and developed a theory of colour based on how prisms split sunlight into a rainbow-coloured spectrum.

**Blaise Pascal** (1623 – 1662) was a French mathematician, physicist and philosopher. He invented some of the first mechanical calculators, as well as working on projective geometry, probability and the physics of the vacuum.

Most famously, Pascal is remembered for naming *Pascal’s Triangle*, an infinite triangle of numbers with some amazing properties.

The English mathematician **John Wallis** (1616 – 1703) contributed to the development of calculus, invented the number line and the symbol ∞ for infinity, and served as chief cryptographer for Parliament and the royal court.

**Pierre de Fermat** (1607 – 1665) was a French mathematician and lawyer. He was an early pioneer of calculus, as well as working in number theory, probability, geometry and optics.

In 1637, he wrote a short note in the margin of one of his textbooks, claiming that the equation *Fermat’s Last Theorem*, and one of the most famous unsolved problems in mathematics – until it was finally proven in 1994.

**Bonaventura Cavalieri** (1598 – 1647) was an Italian mathematician and monk. He developed a precursor to infinitesimal calculus, and is remembered for Cavalieri’s principle to find the volume of solids in geometry.

Cavalieri also worked optics and motion, introduction logarithms to Italy, and exchanged many letters with Galileo Galilei.

**René Descartes** (1596 – 1650) was a French mathematician and philosopher, and one of the key figures in the Scientific Revolution. He refused to accept the authority of previous philosophers, and one of his best-known quotes is “I think, therefore I am”.

Descartes is the father of *analytical geometry*, which allows us to describe geometric shapes using algebra. This was one of the prerequisites, which allowed Newton and Leibnitz to invent *calculus* a few decades later.

He is credited with the first use of superscripts for powers or exponents, and the *cartesian coordinate system* is named after him.

**Girard Desargues** (1591 – 1661) was a French mathematician, engineer, and architect. He designed numerous buildings in Paris and Lyon, helped construct a dam, and invented a mechanism for raising water using epicycloids.

In mathematics, Desargues is considered the father of *projective geometry*. This is a special kind of geometry in which parallel lines meet at at “point at infinity”, the size of shapes does not matter (only their proportions), and all four conic sections (circle, ellipse, parabola and hyperbola) are essentially the same.

**Marin Mersenne** (1588 – 1648) was a French mathematician and priest. Because of the frequent exchanges with his contacts in the scientific world during the 17th century, he has been called the “the post-box of Europe”.

Today we mostly remember him for the *Mersenne primes*, prime numbers that can be written as

**Johannes Kepler** (1571 – 1630) was a German astronomer and mathematician. He was the *imperial mathematician* in Prague, and he is best known for his three *laws of planetary motion*. Kepler also worked in optics, and invented an improved telescope for his observations.

**Galileo Galilei** (1564 – 1642) was an Italian astronomer, physicist and engineer. He used one of the first telescopes to make observations of the night sky, where he discovered the four largest moons of Jupiter, the phases of Venus, sunspots, and much more.

Galileo, sometimes called the “father of modern science”, also studied the motion of objects in free fall, kinematics, material science, and invented the thermoscope (an early thermometer).

He was a vocal proponent of *Heliocentrism*, the idea that the Sun was at the centre of our solar system. This eventually led to him being tried by the Catholic Inquisition: Galileo was forced to recant and spent the rest of his life under house arrest.

**John Napier** (1550 – 1617) was a Scottish mathematician, physicist, and astronomer. He invented logarithms, popularised the use of the decimal point, and created “Napier’s bones”, a manual calculating device that helped with multiplication and division.

**Simon Stevin** (1548 – 1620) was Flemish mathematician and engineer. He was one of the first people to use and write about decimal fractions, and made many other contributions to science and engineering.

**François Viète** (1540 – 1603) was a French mathematician, lawyer, and advisor to Kings Henry III and IV of France. He made significant advances in Algebra, and first introduced the use of letters to represent variables.

Viète discovered the connection between the roots and coefficients of a polynomial, called *Viète's formula*. He also wrote books about geometry and trigonometry, including calculating *π* to 10 decimal places using a polygon with 393216 sides.

The Italian **Gerolamo Cardano** (1501 – 1576) was one of the most influential mathematicians and scientists of the Renaissance. He investigated hypercycloids, published Tartaglia’s and Ferrari’s solution for cubic and quartic equations, was the first European to systematically use negative numbers, and even acknowledged the existing of imaginary numbers (based on

Cardano also made some early progress in probability theory and introduced binomial coefficients and binomial theorem to Europe. He invented many mechanical devices, including combination locks, gyroscopes with three degrees of freedom, and drive shafts (or Cardan shafts) that are still used in vehicles today.

**Niccolò Fontana Tartaglia** (1499 – 1557) was an Italian mathematician, engineer and bookkeeper. He published the first Italian translations of Archimedes and Euclid, found a formula for solving any cubic equation (including the first real application of complex numbers), and used mathematics to investigate the projectile motion of cannonballs.

**Nicolaus Copernicus** (1473 – 1543) was a Polish mathematician, astronomer and lawyer. During his life, most people believed in the *Geocentric* model of the universe, with Earth at the centre and everything else rotating around it.

Copernicus created a new model, where the sun is at the centre, and Earth moves around it on a circle. He also predicted that Earth rotates around its axis once every day. Afraid that it would upset the Catholic church, he only published the model just before his death – triggering what is now called the *Copernican Revolution*.

Copernicus also worked as a diplomat and physician, and made important contributions to economist.

**Leonardo da Vinci** (1452 – 1519) was an Italian artist and polymath. His interests ranged from painting, sculpting and architecture to engineering, mathematics, anatomy, astronomy, botany and cartography. He is often seen as the prime example of a “Universal Genius” and was one of the most diversely talented individuals ever to have lived.

Leonardo was born in Vinci, educated in Florence, and worked in Milan, Rome, Bologna, and Venice. Only 15 of his paintings have survived, but among them are some of the best known and most reproduced works in the world, including the *Mona Lisa* and *The Last Supper*.

His notebooks contain a vast number of drawings, inventions, and scientific diagrams – including the first flying machines and helicopters, hydraulic pumps, bridges, and much more.

**Luca Pacioli** was an influential Italian friar and mathematician, who invented the standard symbols for plus and minus (+ and –). He was one of the first accountants in Europe, where he introduced double-entry book-keeping. Pacioli collaborated with Leonardo da Vinci, and also wrote about arithmetic and geometry.

**Johann Müller Regiomontanus** (1436 – 1476) was a German mathematician and astronomer. He made great advances in both fields, including creating detailed astronomical tables and publishing multiple textbooks.

**Madhava of Sangamagramma** (c. 1340 – 1425) was a mathematician and astronomer from southern India. All of his original work has been lost, but he had a great impact on the development of mathematics.

Madhava first used infinite series to approximate trigonometric functions, which was a significant step towards the development of calculus many centuries later. He also studied geometry and algebra, and found an exact formula for *π* (also using infinite series).

**Nicole Oresme** (c. 1323 – 1382) was an important French mathematician, philosopher and bishop, living in the late Middle Ages. He invented coordinate geometry, long before Descartes, was the first to use fractional exponents, and worked on infinite series. He wrote about economics, physics, astronomy and theology, and was an advisor to King Charles V of France.

**Zhu Shijie** (朱世杰, 1249 – 1314) was one of the greatest Chinese mathematicians. In is book *Jade Mirror of the Four Unknowns*, he showed how to solve 288 different problem using systems of polynomial equations and four variables (called *Heaven*, *Earth*, *Man* and *Matter*).

Zhu made extensive use of Pascal’s triangle. He also invented rules for solving systems of linear equations – predating our modern matrix methods by many centuries.

**Qin Jiushao** (秦九韶, c. 1202 – 1261) was a Chinese mathematician, inventor and politician. In his book *Shùshū Jiǔzhāng*, he published numerous mathematical discoveries, including the important *Chinese remainder theorem*, and wrote about surveying, meteorology and the military.

Qin first developed a method for numerically solving polynomial equations, which is now known as *Horner’s method*. He found a formula for the area of a triangle based on the length of its three sides, calculated the sum of arithmetic series, and introduced a symbol for “zero” into Chinese mathematics.

Qin also invented *Tianchi basins*, which were used to measure rainfall and gather meteorological data important for farming.

**Leonardo Pisano**, commonly known as **Fibonacci** (1175 – 1250) was an Italian mathematician. He is best known for the number sequence named after him: 1, 1, 2, 3, 5, 8, 13, …

Fibonacci is also responsible for popularising the Arabic numerals (0, 1, 2, 3, 4, …) in Europe, which was still using Roman numerals (I, V, X, D, …) in the 12th century AD. He explained the decimal system in a book called “Liber Abaci”, a practical textbook for merchants.

**Bhaskara II** (1114 – 1185) was an Indian mathematician and astronomer. He discovered some of the basic concepts of calculus, more than 500 years before Leibnitz and Newton. Bhaskara also established that division by zero yields infinity, and solved various quadratic, cubic, quartic and Diophantine equations.

**Omar Khayyam** (عمر خیّام, 1048 – 1131) was a Persian mathematician, astronomer and poet. He managed to classify and solve all cubic equations, and found new ways to understand Euclid’s *parallel axiom*. Khayyam also designed the *Jalali calendar*, a precise solar calendar that is still used in some countries.

**Hasan Ibn al-Haytham** (أبو علي، الحسن بن الحسن بن الهيثم, c. 965 – 1050) lived in Cairo during the Islamic Golden Age, and studied mathematics, physics, astronomy, philosophy and medicine. He was a proponent of the *scientific method*: the belief that any a scientific hypothesis must be verified using experiments or mathematical logic – centuries before European scientists during the Renaissance.

Al-Haytham was particularly interested in optics and visual perception. He also derived a formula for the sum of fourth powers (

**Muhammad Al-Karaji** (ابو بکر محمد بن الحسن الکرجی, c. 953 – 1029) was a Persian mathematician and engineer. He was the first person to use *prove by induction*, which allowed him to prove the binomial theorem.

The Persian mathematician **Muhammad Al-Khwarizmi** (محمد بن موسى الخوارزمي, 780 – 850) lived during the golden age of the Muslim Abbasid regime in Baghdad. He worked at the “House of Wisdom”, which contained the first large collection of academic books since the destruction of the Library of Alexandria.

Al-Khwarizmi has been called the “father of algebra” – in fact, the word *algebra* comes from the Arabic title of his most important book: “The Compendious Book on Calculation by Completion and Balancing”. In it, he showed how to solve linear and quadratic equations, and for many centuries, it was the main mathematics textbook at European universities.

Al-Khwarizmi also worked in astronomy and geography, and the word “algorithm” is named after him.

The Indian mathematician **Brahmagupta** (c. 598 – 668 AD) invented the rules for addition, subtraction and multiplication with zero and negative numbers. He was also an astronomer and made many other discoveries in mathematics. Unfortunately, his writings did not contain any proofs, so we don’t know how he derived his results.

**Aryabhata** (आर्यभट) was one of the first mathematicians and astronomers in the golden age of Indian mathematics. He defined trigonometric functions, solved simultaneous quadratic equations, found approximations for *π*, and realised that *π* is irrational.

**Hypatia** (c. 360 – 415 AD) was a prominent astronomer and mathematician in ancient Alexandria. She was also the first female mathematician whose life and work are reasonably well recorded. She edited or wrote commentaries on many of the scientific books of her time, and constructed astrolabes and hydrometers.

She was renowned during her life as a great teacher, and she advised Orestes, the Roman prefect of Alexandria. Orestes’ feud with Cyril, the bishop of Alexandria, led to Hypadia being murdered by a mob of Christians.

**Diophantus** was a Hellenistic mathematician who lived in Alexandria. Most of his works are about solving polynomial equations with several unknowns. These are now called *Diophantine equation* and remain an important area of research today.

It was while reading one of Diophantus’ books, many centuries later, that *Pierre de Fermat* proposed one of these equations had no solution. This became known as “Fermat’s Last Theorem”, and was only solved in 1994.

**Claudius Ptolemy** (c. 100 – 170 AD) was a Greco-Roman mathematician, astronomer, geographer and astrologer. He is best remembered for the *Ptolemaic* or *Geocentric* model of our universe – that Earth is in the centre and all planets and the sun revolve around this.

While we know today this model is incorrect, Ptolemy’s scientific impact is indisputable. He developed trigonometric tables with many practical applications, which remained the most accurate for many centuries. He also created detailed maps of the Earth, and wrote about music theory and optics.

**Nicomachus of Gerasa** (c. 60 – 120) was an ancient Greek mathematician who also spent much time thinking about the mystical properties of numbers. His book *Introduction to Arithmetic* contains the first mention of perfect numbers.

**Apollonius of Perga** (c. 200 BC) was a Greek mathematician and astronomer best known for his work on the four *conic sections*.

**Eratosthenes of Cyrene** (c. 276 BC – 195 BC) was a Greek mathematician, geographer, astronomer, historian, and poet. He spent much of his life in Egypt, as head of the library of Alexandria.

Among many other achievements, Eratosthenes calculated the circumference of the Earth, measured the tilt of the Earth’s axis of rotation, estimated the distance to the sun, and created some of the first maps of the world.

He also invented the “Sieve of Eratosthenes”, an efficient way to calculate *prime numbers*.

**Archimedes** (c. 287 – c. 212 BC) was an ancient Greek scientist and engineer, and one of the greatest mathematicians of all time. He discovered many concepts of calculus and worked in geometry, analysis and mechanics.

While taking a bath, Archimedes discovered a way to determine the volume of irregular objects using the amount of water they displaced when submerged. He was so excited by this discovery that he ran out on the street, still undressed, yelling *“Eureka!”* (Greek for *“I have found it!”*).

As an engineer, he built ingenious defence machines during the siege of his home city Syracuse in Sicily. After two years, the Romans finally managed to enter, and Archimedes was killed. His last words were *“Do not disturb my circles”* – which he was studying at the time.

**Pingala** (पिङ्गल) was an ancient Indian poet and mathematician who lived around 300BC, but very little is known about his life. He wrote the Chandaḥśāstra, where he analysed Sanskrit poetry mathematically. It also contained the first known explanations of binary numbers, Fibonacci numbers and Pascal’s triangle.

**Euclid of Alexandria** (around 300 BC) was a Greek mathematician and is often called the *father of geometry*. He published a book *Elements* that first introduced Euclidean geometry and contained many important proofs in geometry and number theory. It was the main mathematics textbook until the 19th century. He taught mathematics in Alexandria, but very little else is known about his life.

**Aristotle** (Ἀριστοτέλης, c. 384 – 322 BC) was a philosopher in Ancient Greece. Together with his teacher *Plato*, he is considered the “Father of Western Philosophy”. He was also the private tutor of Alexander the Great.

Aristotle wrote about science, mathematics, philosophy, poetry, music, politics, rhetoric, linguistics, and many other subjects. His work was highly influential during the Middle Ages and into the Renaissance, and his views on ethics and other philosophical questions are still being discussed today.

Aristotle is also the first known person to formally study *logic*, including its applications in science and mathematics.

**Plato** (c. 425 – 347 BC) was a philosopher in ancient Greece, and – together with his teacher Socrates and his student Aristotle – laid the very foundation of Western philosophy and science.

Plato founded the Academy of Athens, the first higher learning institution in the Western world. His many writings on philosophy and theology, science and mathematics, politics and justice, make him one of the most influential thinkers of all time.

The Greek mathematician **Democritus** (c. 460 – 370 BC), may be the first person to speculate that all matter was made up of tiny *atoms* and is considered the “father of modern science”. He also made many discoveries in geometry, including the formula for the volume of prisms and cones.

**Pythagoras of Samos** (c. 570 – 495 BC) was a Greek philosopher and mathematician. He is best known for proving *Pythagoras’ Theorem*, but made many other mathematical and scientific discoveries.

Pythagoras tried to explain music in a mathematical way, and discovered that two tones sound “nice” together (consonant) if the ratio of their frequencies is a simple fraction.

He also founded a school in Italy where he and his students worshipped mathematics almost like a religion, while following a number of bizarre rules – but the school was eventually burned down by their adversaries.

**Thales of Miletus** (c. 624 – 546 BC) was a Greek mathematician and philosopher.

Thales is often recognised as the first scientist in Western civilisation: rather than using religion or mythology, he tried to explain natural phenomena using a scientific approach. He is also the first individual in history that has a mathematical discovery named after him: Thales’ theorem.