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Great Thinkers, Great Theorems

Delve into the mechanics of some of math's greatest and most awe-inspiring achievements.
Great Thinkers, Great Theorems is rated 4.8 out of 5 by 111.
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Rated 5 out of 5 by from Great Professor, Great Lectures I wasn't sure how much I was going to get into this -- my high school and college math are decades in the past, didn't know if this was going to be over my head or not. But I was instantly drawn in by the professor and not only his expertise in math, but with the way he could tell a story. I learned more about the past and the history and the people than I ever knew before, and I found myself replaying lectures because they were so interesting. Some of the exercises were over my head (although much of it came back as I worked through them). But I was intrigued and pulled into it enough that I actually bought or checked out from the library several of the resource books listed. He woke up an interest that has continued well past the lectures. Professor Dunham was so engaging, so entertaining and so enthusiastic about the people and the theorems that I recall thinking that if my own professors had been half as interesting as him, I would have had a much easier time in college. I'm sure I will be returning to this course again!
Date published: 2022-10-16
Rated 5 out of 5 by from Outstanding Course Truly an exceptional educational experience. Prof. Dunham brings an engaging, enthusiastic manner that is infectious. His inclusion of the “Mount Rushmore” of mathematicians is on the mark, as is his selection of their works to address. Many of the proofs he address are quite complex but he goes through them systematically, assisted by outstanding graphics from the production team. Only criticism I can offer is that I was left wanting more! Perhaps 20th century mathematicians, say Kurt Godel and the Incompleteness theorems, etc. (Note: reviewer is a Ph.D. Experimental Astrophysicist who hasn’t been in a classroom in many years.)
Date published: 2022-07-27
Rated 5 out of 5 by from Excellent course Presented in a interesting way 20 characters too short20 characters too short way
Date published: 2022-01-20
Rated 5 out of 5 by from Great course! I really enjoyed this course. A beautiful mixture of mathematical history and amazing (but easy to follow) demonstrations. The instructor's excitement about the subject matter is contagious!
Date published: 2021-11-14
Rated 5 out of 5 by from Fabulous! This course was superb! Professor Dunham covers the landscape of mathematical thought from antiquity through the late 19th century with humor and interesting stories about the great mathematicians while at the same time weaving in some terrific mathematics. it provides a remarkable appreciation for the genius of generations of famous mathematicians, many of whom one has heard, and some of whom are more obscure. I can't wait to watch the entire course again.
Date published: 2021-08-29
Rated 5 out of 5 by from Excellent course, excellent lecturer. I found this to be an excellent course. Professor Dunham is a warm and likeable lecturer, with a pleasant voice and always good-humoured. His preparation and delivery was excellent - obviously achieved with much practice. He spoke clearly and takes pains to make sure everything is clearly understandable. (Towards the end of the course I actually switched to 1.5x speed, as he was clear even at this speed). It didn't feel like I was being lectured to - it felt like I was being entertained! I especially like his particular way of adding little fillers to keep contact with his audience - "almost there", "let's recall what we are trying to prove", etc. Most importantly, the content was very good. It is an entertaining history of mathematics, from 1850 BC to the end of the 19th Century, from the Ancient Greeks to Cantor, with lots of good stories to illustrate the lives and circumstances of the great mathematicians that Professor Dunham has selected (and a good choice it is too!). He included just the right amount of mathematics for me - enough for me to work along and understand the theorems better, but making this a pleasure rather than an effort. I do have a Maths degree, but still learned a lot, especially about how mathematics developed through the centuries. For example, I didn't realise that the use of x's and y's in algebra was quite a late invention - until the time of Newton, problems and solutions were communicated using long-winded prose! The presentation is a monologue - but with on-screen animated graphics which are all very clear and helpful. Overally, I would recommend this course to anyone (adult or teenager onwards) with any slight interest in mathematics, at any level of mathematical education.
Date published: 2021-08-22
Rated 4 out of 5 by from Great thinkers, Great theorems. Only up to Lecture 7 but very pleased so far. Impressed by presentation, particularly the logical process.
Date published: 2021-08-09
Rated 5 out of 5 by from Outstanding Course Professor William Durham has done a masterful job in presenting the great theorems and major works of the world's most famous and creative mathematicians. He has taken a complex subject and framed it is such a way that it is clear, entertaining and provides an understanding of this beautiful, abstract science. Through the use of theorems, he shows the genius and creative power of these gifted individuals. One can appreciate why Professor Durham is an award winning teacher.
Date published: 2021-08-03
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Explore the most awe-inspiring theorems in the 3,000-year history of mathematics with the 24 lectures of Great Thinkers, Great Theorems. Professor William Dunham, an award-winning teacher with a talent for conveying the essence of mathematical ideas, reveals how great minds like Pythagoras, Newton, and Euler crafted theorems that would revolutionize our understanding of the world. Approaching great theorems the way an art course approaches great art, Professor Dunham will open your mind to new levels of mathematics appreciation.


William Dunham

Mathematics is perhaps the most useful of subjects, but my interests are not utilitarian.  Rather, I look to the great landmarks from the history of mathematics and present them as creative achievements that are a joy to behold.


Muhlenberg College

Dr. William Dunham is the Truman Koehler Professor of Mathematics at Muhlenberg College in Allentown, Pennsylvania. He earned his undergraduate degree from the University of Pittsburgh and his M.S. and Ph.D. in Mathematics from The Ohio State University.  Before his current appointment at Muhlenberg, Dr. Dunham taught at Hanover College in Indiana, receiving teaching awards from both institutions as well as the Award for Distinguished College or University Teaching from the Eastern Pennsylvania and Delaware Section of the Mathematical Association of America. He was a Visiting Professor at Ohio State and at Harvard University, where he was invited to teach an undergraduate course on the work of Leonhard Euler and to deliver the Clay Public Lecture in 2008.  Dr. Dunham's great theorems approach to teaching mathematics was fostered by a 1983 summer grant from the Lilly Endowment, which also led to his first book, Journey Through Genius: The Great Theorems of Mathematics—a Book-of-the-Month Club selection that has been translated into five languages. Other books followed in addition to articles on mathematics and its history, earning him numerous awards from the Mathematical Association of America and other organizations. He has presented popular talks on mathematics throughout the United States and has appeared on the BBC and NPR's Talk of the Nation: Science Friday.

By This Professor

Theorems as Masterpieces

01: Theorems as Masterpieces

Certain theorems stand out as great masterpieces of mathematics that can be appreciated as great works of art. After hearing Professor Dunham explain this approach, discover the two ways of proving a theorem: direct proof and indirect proof. Also, meet some of the great thinkers whose ideas you will be studying.

32 min
Mathematics before Euclid

02: Mathematics before Euclid

Investigate three non-Greek civilizations that had robust traditions in mathematics. Then encounter a pair of Greek mathematicians who predated Euclid, but who left very deep footprints: Thales and Pythagoras—the latter renowned for the theorem that bears his name.

31 min
The Greatest Mathematics Book of All

03: The Greatest Mathematics Book of All

Begin your exploration of the work widely considered the greatest mathematical text of all time: Euclid's Elements. Discover why these 13 succinct books have been so influential for so long as you delve into the ground-laying definitions, postulates, common notions, and theorems from book I.

29 min
Euclid's Elements-Triangles and Polygons

04: Euclid's Elements-Triangles and Polygons

Continuing your journey through Euclid, work your way toward his most famous result: his proof of the Pythagorean theorem—a demonstration of remarkable visual and intellectual beauty. Also, cover some of the techniques from book IV for constructing regular polygons.

32 min
Number Theory in Euclid

05: Number Theory in Euclid

In addition to being a geometer, Euclid was a pioneering number theorist, a subject he took up in books VII, VIII, and IX of the Elements. Focus on his proof that there are infinitely many prime numbers, which Professor Dunham considers one of the greatest proofs in all of mathematics.

29 min
The Life and Works of Archimedes

06: The Life and Works of Archimedes

Even more distinguished than Euclid was Archimedes, whose brilliant ideas took centuries to fully absorb. Probe the life and famous death of this absent-minded thinker, who once ran unclothed through the streets, shouting "Eureka!" ("I have found it!") on solving a problem in his bath.

29 min
Archimedes' Determination of Circular Area

07: Archimedes' Determination of Circular Area

See Archimedes in action by following his solution to the problem of determining circular area—a question that seems trivial today but only because he solved it so simply and decisively. His unusual strategy relied on a pair of indirect proofs.

32 min
Heron's Formula for Triangular Area

08: Heron's Formula for Triangular Area

Heron of Alexandria (also called Hero) is known as the inventor of a proto-steam engine many centuries before the Industrial Revolution. Discover that he was also a great mathematician who devised a curious method for determining the area of a triangle from the lengths of its three sides.

31 min
Al-Khwarizmi and Islamic Mathematics

09: Al-Khwarizmi and Islamic Mathematics

With the decline of classical civilization in the West, the focus of mathematical activity shifted to the Islamic world. Investigate the proofs of the mathematician whose name gives us our term "algorithm": al-Khwarizmi. His great book on equation solving also led to the term "algebra."

30 min
A Horatio Algebra Story

10: A Horatio Algebra Story

Visit the ruthless world of 16th-century Italian universities, where mathematicians kept their discoveries to themselves so they could win public competitions against their rivals. Meet one of the most colorful of these figures: Gerolamo Cardano, who solved several key problems. In secret, of course.

29 min
To the Cubic and Beyond

11: To the Cubic and Beyond

Trace Cardano's path to his greatest triumph: the solution to the cubic equation, widely considered impossible at the time. His protégé, Ludovico Ferrari, then solved the quartic equation. Norwegian mathematician Niels Abel later showed that no general solutions are possible for fifth- or higher-degree equations.

32 min
The Heroic Century

12: The Heroic Century

The 17th century saw the pace of mathematical innovations accelerate, not least in the introduction of more streamlined notation. Survey the revolutionary thinkers of this period, including John Napier, Henry Briggs, René Descartes, Blaise Pascal, and Pierre de Fermat, whose famous "last theorem" would not be proved until 1995.

31 min
The Legacy of Newton

13: The Legacy of Newton

Explore the eventful life of Isaac Newton, one of the greatest geniuses of all time. Obsessive in his search for answers to questions from optics to alchemy to theology, he made his biggest mark in mathematics and science, inventing calculus and discovering the law of universal gravitation.

30 min
Newton's Infinite Series

14: Newton's Infinite Series

Start with the binomial expansion, then turn to Newton's innovation of using fractional and negative exponents to calculate roots—an example of his creative use of infinite series. Also see how infinite series allowed Newton to approximate sine values with extraordinary accuracy.

31 min
Newton's Proof of Heron's Formula

15: Newton's Proof of Heron's Formula

Return to Heron's ancient formula for determining the area of a triangle to consider Newton's proof using algebraic techniques—an approach he also applied to other geometry problems. The steps are circuitous, but the result bears Newton's stamp of genius.

32 min
The Legacy of Leibniz

16: The Legacy of Leibniz

Probe the career of Newton's great rival, Gottfried Wilhelm Leibniz, who came relatively late to mathematics, plunging in during a diplomatic assignment to Paris. In short order, he discovered the "Leibniz series" to represent π, and within a few years he invented calculus independently of Newton.

31 min
The Bernoullis and the Calculus Wars

17: The Bernoullis and the Calculus Wars

Follow the bitter dispute between Newton and Leibniz over priority in the development of calculus. Also encounter the Swiss brothers Jakob and Johann Bernoulli, enthusiastic supporters of Leibniz. Their fierce sibling rivalry extended to their competition to outdo each other in mathematical discoveries.

31 min
Euler, the Master

18: Euler, the Master

Meet history's most prolific mathematician, Leonhard Euler, who went blind in his sixties but kept turning out brilliant papers. A sampling of his achievements: the number e, crucial in calculus; Euler's identity, responsible for the most beautiful theorem ever; Euler's polyhedral formula; and Euler's path.

30 min
Euler's Extraordinary Sum

19: Euler's Extraordinary Sum

Euler won his spurs as a great mathematician by finding the value of a converging infinite series that had stumped the Bernoulli brothers and everyone else who tried it. Pursue Euler's analysis through the twists and turns that led to a brilliantly simple answer.

31 min
Euler and the Partitioning of Numbers

20: Euler and the Partitioning of Numbers

Investigate Euler's contribution to number theory by first warming up with the concept of amicable numbers—a truly rare breed of integers until Euler vastly increased the supply. Then move on to Euler's daring proof of a partitioning property of whole numbers.

31 min
Gauss-the Prince of Mathematicians

21: Gauss-the Prince of Mathematicians

Dubbed the Prince of Mathematicians by the end of his career, Carl Friedrich Gauss was already making major contributions by his teen years. Survey his many achievements in mathematics and other fields, focusing on his proof that a regular 17-sided polygon can be constructed with compass and straightedge alone.

30 min
The 19th Century-Rigor and Liberation

22: The 19th Century-Rigor and Liberation

Delve into some of the important trends of 19th-century mathematics: a quest for rigor in securing the foundations of calculus; the liberation from the physical sciences, embodied by non-Euclidean geometry; and the first significant steps toward opening the field to women.

31 min
Cantor and the Infinite

23: Cantor and the Infinite

Another turning point of 19th-century mathematics was an increasing level of abstraction, notably in the approach to the infinite taken by Georg Cantor. Explore the paradoxes of the "completed" infinite, and how Cantor resolved this mystery with transfinite numbers, exemplified by the transfinite cardinal aleph-naught.

29 min
Beyond the Infinite

24: Beyond the Infinite

See how it's possible to build an infinite set that's bigger than the set of all whole numbers, which is itself infinite. Conclude the course with Cantor's theorem that the transcendental numbers greatly outnumber the seemingly more abundant algebraic numbers—a final example of the elegance, economy, and surprise of a mathematical masterpiece.

31 min