Formulas, Techniques, and Methods in Mathematics: Finding the People Behind the Numbers
The Pythagorean Theorem, Newton’s laws, and calculus are all mathematical
terms that most of us are familiar with, but the history and names behind these
terms are often forgotten. Many early mathematicians made genius contributions
to the field, and without their knowledge, math would be a completely different
discipline. Numerous advances in mathematics came from collaboration and small contributions
from many mathematicians over long periods of time. The following are descriptions
of some of the major contributors to the field of mathematics with external links
to further information on their personal lifves as well as the mathematic technique
they developed or helped further in the field.
Pythagoras (ca. 570 - ca. 490 B.C.)
One of the most famous names in mathematics, Pythagoras developed the core theorem
in trigonometry, the Pythagorean Theorem. Pythagoras was a Greek mathematician and
also a philosopher, who founded the religious movement Pythagoreanism, which combined
metaphysical beliefs with mathematical knowledge. Although his famous theorem was
generally accepted to be true about two hundred years before his birth by the Sumerians,
Pythagoras is given credit for proving that it is true. There is little known about
Pythagoras’s early life, and what is known is often fictionalized, since there
were few reliable sources written about him during his lifetime.
Greek Philosophers: Pythagoras
– A summary of the life of Pythagoras as a mathematician and philosopher.
and the Pythagoreans – A history of the Pythagorean School and the
importance of religion and mathematics to Pythagoras's school and followers.
- A philosophical look at Pythagoras’s mathematical and metaphysical idea,
the Tetraktys, a geometric figure made out of an equilateral triangle.
The Pythagorean Theorem – A proof of the Pythagorean Theorem, as well
as an introduction to Pythagorean triples.
Animated Proof of the Pythagorean
Theorem – An animation that illustrates a proof of the formula, c2
= a2 + b2 .
Euclid (ca. 325 - ca.220 B.C.)
Like Pythagoras, Euclid is another early Greek mathematician who lacks much of a
written record of his life. Based on a few mentions of Euclid in written records
from Greece, it is believed that he studied at Plato’s Academy in Athens.
Euclid made great contributions to the field of geometry through his book The Elements.
It was a popular geometry textbook up until the early twentieth century. In The
Elements, Euclid describes the principles of what is today known as Euclidian
Summary of Euclid – A
synopsis of Euclid’s life and a listing of his ten axioms or postulates of
Biography of Euclid
– A short biography of what is known about Euclid’s life and his works.
– Definitions, postulates, and propositions from Euclid’s work The Elements.
Interview with Euclid – An interview with Euclid written in first
person question and answer format that reveals biographical information.
Euclidian Geometry – Explanations of postulates from Euclid’s
The Elements which form the basis of Euclidean Geometry.
Leonardo Fibonacci (1170-1250)
Fibonacci was known by many names including Leonardo of Pisa, Leonardo Pisano Bigollo,
and Leonardo Bonacci, but his surname Fibonacci has stuck with him due to his namesake
discovery, the Fibonacci sequence. Fibonacci was an Italian-born mathematician famous
for his spread of the Hindu-Arabic numeral system throughout Europe because of his
writings and the Fibonacci sequence. As a young man, he traveled with his merchant
father and found that the Hindu-Arabic numeral system was more efficient than the
Roman numeral system. In his book, the Liber Abaci , Fibonacci shows why
this number system is more efficient and also introduces his famous sequence as
a problem involving the growth of the rabbit population. The Fibonacci sequence
is easily viewable in nature and everyday life and has become a popular topic in
movies, novels, and art.
Was Fibonacci? – A biography of Fibonacci and summary of his mathematical
Fibonacci Numbers Spelled Out
– Different derivations of the Fibonacci sequence mathematically spelled out.
in Nature – Pictures and diagram of examples of the Fibonacci sequence
The Fibonacci Association –
This association’s website, named after Fibonacci, contains information on
the Fibonacci numbers, Number Theory, and links to art displaying the Fibonacci
sequence in the spiral form.
The Golden Ratio –
An explanation of the Golden Ratio which is demonstrated in Fibonacci’s sequence.
The Fibonacci Sequence
Written – Shows how the Fibonacci Sequence can be written as a rule
and how to use the golden ratio to calculate Fibonacci numbers.
Pierre De Fermat (1601-1665)
Fermat was born in France and became a lawyer as a young man. After he received
his degree in civil law, he spent the remainder of his life working as the councillor
at the High Court of Judicature in Toulouse. Although he maintained the status
of “amateur mathematician” his work led to developments in infinitesimal
calculus, made contributions to Number Theory, analytic geometry, and the adequality
technique. Fermat also worked with Blaise Pascal, with whom he had a close relationship,
to discover the theory of probability. Fermat left his notable theorem, Fermat’s
Last Theorem unproven, and findng the proof of this theorem became the ultimate
goal of many mathematicians. Finally in the late twentieth century, mathematician
Andrew Wiles was able to write the proof of Fermat's theorem.
of Pierre de Fermat – A summary of Fermat’s personal life as
well as the role he played in mathematics.
What is the Last Theorem – A description with diagrams of what Fermat’s
Last Theorem entails.
Fermat’s Last Theorem
– This contains a lengthy description of Fermat’s theorem, a shortened
proof of the theorem, and details about the race to find the proof.
Andrew Wiles – Princeton’s faculty profile of the mathematician
who was able to prove Fermat’s Last Theorem.
Pierre de Fermat –
Biography of Fermat with diagrams of some of his mathematical concepts.
Blaise Pascal (1623-1662)
Pascal was a French mathematician, philosopher, inventor, and physicist. His father
was a tax commissioner, which prompted Pascal to invent a calculating machine as
a young man. He was a follower of Jansenism, a movement within Catholicism. His
primary contributions to mathematics were Pascal’s triangle and his collaboration
with Fermat on the theory of probability. Due to Pascal’s deep religious beliefs,
in 1654, he stopped all his work in mathematics, but broke this constrainment a
few years later when he offered up a competition to see who could find the numerical
derivation of a cycloid; under a pseudonym, he submitted the winning answer. In
honor of Pascal’s contributions to math and science, Pascal’s law, Pascal,
the unit of pressure, and the programming language also referred to by his given
surname were named so for his accomplishments.
– A summary and timeline of major events in Blaise Pascal’s life.
– The European Graduate Schools’ biography of Pascal.
– This page gives the history, construction, patterns, and applications of
The Fibonacci sequence in Pascal’s
Triangle – A diagram that shows how the Fibonacci sequence appears
in Pascal’s Triangle.
The Cycloid (PDF) – Information
about the cycloid and the math behind it.
Sir Issac Newton (1643-1727)
Newton was engaged in a plethora of different scientific fields including optics,
mathematics, mechanics, and gravitation, and his famous laws are now prevelently
used in many scientific and mathematical subjects. Newton was born in England. As
a boy, his mother wished for him to become a farmer, but he was able to go on with
his schooling and eventually graduated from Trinity College in Cambridge. Newton’s
famous three laws of motion formularized inertia, applied force and momentum, and
acceleration. Often illustrated in popular culture, Newton himself spread the idea
that he was able to formulate the law of gravitation after an apple fell on his
head from the branch of a tree overhead; although it is believed it didn’t
happen quite this way since it took Newton about twenty years to fully write the
theory. The unit of force, the newton, is named for Newton’s contribution
of the first and second laws of motion having to do with force.
Sir Issac Newton – A biography of Newton with a summary of his achievements.
The Mind of Issac
Newton – A Multimedia project from McMaster University with information
on Newton’s innovations.
of Motion – This provides a simplified explanation of Newton’s
three laws of motion.
Newton: The Universal Laws of Gravitation – An explanation of how
Newton came about discovering the universal laws of gravitation and the math behind
Issac Newton’s Life –
Information on Issac Newton’s life and the fields he made breakthroughs in.
Leonhard Euler (1701-1783)
Considered by many to be one of the greatest mathematicians of all times, Swiss-born
Leonhard Euler made advancements in infinitesimal calculus, graph theory, and is
notable for much of the modern day notation that is currently used in mathematics.
Euler also made contributions to physics and astronomy. Working with his mathematician
friend Daniel Bernoulli, he helped develop the Euler-Bernoulli beam equation. Throughout
his lifetime, Euler wrote a tremendous amount of books on mathematics. He is immortalized
on postage stamps in Germany, Switzerland, and Russia, and his picture was even
featured on a series of the Swiss banknote.
Leonhard Euler – A
summary of Leohard Euler's life and his contributions to mathematics and science.
Method – Explanations and examples of Euler’s method in
Euler’s Method - Formulas – Information about Euler’s
method and how it is used. Also contains a summary of Euler’s method in formulas.
– An explanation of Euler’s equation which shows the relationship between
the trigonometric function and complex exponential function.
– Details about Euler’s identity and a corollary of the identity and
how they can be used.
Jean Baptiste Joseph Fourier (1768-1830)
Fourier was a French-born mathematician and physicist. At a young age he became
an orphan, but was still able to become obtain a commendable education. He spent
a portion of his life in Egypt as governor after venturing on an expedition with
Napoleon Bonaparte. Fourier performed many experiments on the propagation of heat.
Today he is known for discovering the “greenhouse effect.” Fourier worked
on determinate equations, but never finished them before his death. His work was
later finished by a few other mathematicians, and today Fourier analysis is named
after his original work.
Baptiste Joseph Fourier – A short biography of the famous mathematician.
Fourier Series Demonstrated –
A java applet that demonstrates the Fourier series.
Fourier Series Tutorials – Listings
of interactive flash programs that can help one learn all about Fourier series.
Fourier Transform Table
(PDF) – A table of the Fourier transform.
Heating by the
Greenhouse Effect – An introduction to the greenhouse effect with
a description of how Fourier was able to discover this idea.
Carl Friedrich Gauss (1777-1855)
Gauss was born in Germany and was a very precocious child. He made his first major
mathematical discoveries in his teenage years. Carl Friedrich Gauss led the life
of a perfectionist and published little of his mathematical material due to this
personality trait. His famous Gaussian distribution is more familiarly known today
as the bell curve with which teachers often base their grading systems around. Gauss
also collaborated with the physicist, Wilhelm Weber. Their work led to advancements
in magnetism, and together they invented the electromagnetic telegraph. Around the
time of their collaborative work, Gauss also formulated Gauss’s Law, which
later became one of the four laws of Maxwell’s equations, the foundations
of all modern-day electrical technologies. His legacy has been commemorated in many
forms such as in his image on stamps and currency, a crater on the Moon named Gauss,
and an observation tower in Germany named the Gauss Tower.
Johann Carl Friedrich Gauss
– A summary of Gauss’s life and short timeline of major highlights in
Gauss and His Life Works - A biography of Gauss’s
life with pictures and further links to more information on the mathematician.
Carl Friedrich Gauss
– Information on his major achievements and pictures of stamps and currency
with his picture dedicated to his honor.
Function – The math behind the Gaussian distribution function.
The Normal Distribution
– Characteristics of the Gaussian distribution function, or the more commonly
named bell curve, and an example of one of its applications.
Equations – An examination of Maxwell’s equation which shows
Gauss’s law composing part of the equations.