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.

Pythagoras and the Pythagoreans – A history of the Pythagorean School and the importance of religion and mathematics to Pythagoras's school and followers.

The Tetraktys - 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 geometry.

Summary of Euclid – A synopsis of Euclid’s life and a listing of his ten axioms or postulates of mathematics.

Biography of Euclid – A short biography of what is known about Euclid’s life and his works.

Euclid’s Elements – 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.

Who Was Fibonacci? – A biography of Fibonacci and summary of his mathematical achievements.

Fibonacci Numbers Spelled Out – Different derivations of the Fibonacci sequence mathematically spelled out.

Fibonacci Numbers in Nature – Pictures and diagram of examples of the Fibonacci sequence in nature.

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.

Biography 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.

Blaise Pascal – A summary and timeline of major events in Blaise Pascal’s life.

Pascal’s Biography – The European Graduate Schools’ biography of Pascal.

Pascal’s Triangle – This page gives the history, construction, patterns, and applications of Pascal’s Triangle.

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.

Newton’s Laws of Motion – This provides a simplified explanation of Newton’s three laws of motion.

Sir Issac Newton: The Universal Laws of Gravitation – An explanation of how Newton came about discovering the universal laws of gravitation and the math behind the laws.

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.

Euler’s Method – Explanations and examples of Euler’s method in mathematics.

Euler’s Method - Formulas – Information about Euler’s method and how it is used. Also contains a summary of Euler’s method in formulas.

Euler’s Equation – An explanation of Euler’s equation which shows the relationship between the trigonometric function and complex exponential function.

Euler’s Identity – 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.

Jean 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 his life.

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.

Gaussian Distribution 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.

Maxwell’s Equations – An examination of Maxwell’s equation which shows Gauss’s law composing part of the equations.