Queen of the Sciences: A History of Mathematics

Course No. 1434
Professor David M. Bressoud, Ph.D.
Macalester College
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Course Overview

In the 17th century, the great scientist and mathematician Galileo Galilei noted that the book of nature "cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is not humanly possible to understand a single word of it."

For at least 4,000 years of recorded history, humans have engaged in the study of mathematics. Our progress in this field is a gripping narrative, a never-ending search for hidden patterns in numbers, a philosopher's quest for the ultimate meaning of mathematical relationships, a chronicle of amazing progress in practical fields like engineering and economics, a tale of astonishing scientific discoveries, a fantastic voyage into realms of abstract beauty, and a series of fascinating personal profiles of individuals such as:

  • Archimedes, the greatest of all Greek mathematicians, who met his death in 212 B.C. at the hands of a Roman soldier while he was engrossed in a problem
  • Evariste Galois, whose stormy life in 19th-century radical French politics was cut short by a duel at age 20—but not before he laid the foundations for a new branch of modern algebra called Galois theory
  • Srinivasa Ramanujan, an impoverished college dropout in India who sent his extraordinary equations to the famous English mathematician G. H. Hardy in 1913 and was subsequently recognized as a genius

An inquiring mind is all you need to embark on this supreme intellectual adventure in The Queen of the Sciences: A History of Mathematics, which contains 24 illuminating lectures taught by award-winning Professor of Mathematics David M. Bressoud.

The "Queen of the Sciences"

The history of mathematics concerns one of the most magnificent, surprising, and powerful of all human achievements. In the early 19th century, the noted German mathematician Carl Friedrich Gauss called mathematics the "queen of the sciences" because it was so successful at uncovering the nature of physical reality. Gauss's observation is even more accurate in today's age of quantum physics, string theory, chaos theory, information technology, and other mathematics-intensive disciplines that have transformed the way we understand and deal with the world.

The Queen of the Sciences takes you from ancient Mesopotamia—where the Pythagorean theorem was already in use more than 1,000 years before the Greek thinker Pythagoras traditionally proved it—to the Human Genome Project, which uses sophisticated mathematical techniques to decipher the 3 billion letters of the human genetic code.

Along the way, you meet a remarkable range of individuals whose love of numbers, patterns, and shapes created the grand edifice that is mathematics. These include astrologers, lawyers, a poet, a cult leader, a tax assessor, the author of the most popular textbook ever written, a high school teacher, a blind grandfather, an artist, and several prodigies who died too young.

You find the problems and ideas that preoccupied them can be stated with the utmost simplicity:

  • Is there a method for finding all the prime numbers below a given number? (Eratosthenes, c. 200 B.C.)
  • The equation xn + yn = zn has no whole-number solutions where n is greater than 2. (Pierre de Fermat, 1637)
  • What would it mean if space is non-Euclidean; that is, if it is not flat as described by Euclid? (János Bolyai, 1832)

The second of these propositions, called Fermat's last theorem, is one of the most famous in mathematics. It was followed by this postscript in the book where Fermat jotted it down: "I have a truly marvelous demonstration, which this margin is too narrow to contain." Since Fermat never wrote out his proof, his statement served as a tantalizing challenge to succeeding generations of mathematicians.

The difficult road to a proof of Fermat's last theorem is a theme that surfaces throughout the last half of this course. Among other intriguing facts, you learn that Circle Limit III, a mathematically inspired woodcut by the Dutch artist M. C. Escher, relates directly to the technique that eventually showed the way to a solution by mathematician Andrew Wiles in 1994.

See Mathematics in Context

Professor Bressoud begins the course by defining mathematics as the study of the abstraction of patterns. Mathematics arises from patterns observed in the world, usually patterns expressed in terms of number and spatial relationships. Furthermore, it is a human endeavor found in every culture extending back as far as records go.

The Queen of the Sciences focuses on the European tradition that grew out of early mathematics in Mesopotamia, Egypt, and Greece. The first eight lectures examine these foundations and the contributions of India, China, and the Islamic world, which played important roles in the development of European mathematical achievements. For example:

  • The earliest recorded use of zero as a placeholder was found in a Hindu temple in Cambodia constructed in A.D. 683. Zero had been used a few decades earlier by the Indian astronomer Brahmagupta not as a placeholder but as a number that could be manipulated.
  • An approximation for pi of 355/113 was developed in the 5th century by the Chinese astronomer Zu Chongzhi. Correct to seven decimal places, this approximation would remain the most accurate estimate for more than 1,000 years.
  • The first treatise on al-jabr (restoring) and al-muqabala (comparing)—the process of solving an algebraic equation—was written in A.D. 825 by the Islamic mathematician Abu Jafar al-Kwarizmi. Al-jabr eventually would become the word "algebra" and al-Kwarizmi would become the word "algorithm."

The next eight lectures show how Western Europe, beginning in the late Middle Ages, gathered existing mathematical ideas and refined them into new and powerful tools. The heart of this section is five lectures on the 17th century, when the separate threads of geometry, algebra, and trigonometry began to meld into a cohesive whole, one whose fruits included the creation of calculus by Isaac Newton and Gottfried Wilhelm Leibniz.

Calculus is another recurring theme throughout this course, making its first appearance in the method of exhaustion developed by the ancient Greeks. In the early 17th century, John Napier initiated the idea of logarithms, which added to the examples from which the general rules of calculus emerged. You discover how, in his ceaseless toying with his new invention, Napier chanced on a base that is the equivalent to the modern base of the natural logarithm used in calculus: the famous number now known as e (2.71828 ... ).

After studying the 18th-century contributions of Leonhard Euler—possibly the greatest mathematician who ever lived—you look at how art has influenced geometry and all of mathematics. You investigate mosaics from the Alhambra, prints by M. C. Escher and Albrecht Dürer, and other intriguing shapes and forms.

In the final eight lectures, you explore selected mathematical developments of the past 200 years, including:

  • Joseph Fourier's solution in the early 1800s to the problem of modeling heat flow, which led to a powerful technique called Fourier analysis for making sense of a wide range of complex physical phenomena
  • Bernhard Riemann's new system of geometry in the mid-1800s, which provided a framework for the revolutionary conception of space developed by Albert Einstein in his general theory of relativity
  • Grigori Perelman's recent, startling solution to the Poincaré conjecture proposed by Henri Poincaré in 1904, which earned Perelman the prestigious Fields Medal (which the reclusive Russian mathematician declined)

Learn with an Experienced Teacher

Experienced in teaching mathematics to students of all levels, Professor Bressoud was a Peace Corps volunteer in the West Indies before earning his Ph.D., where he taught mathematics and science to intermediate students. In addition, he has written numerous articles on mathematics education and related issues, including four textbooks that draw heavily on the history of mathematics.

His depth of knowledge and passion for teaching mathematics—which earned him the Mathematical Association of America's Allegheny Mountain Section Distinguished Teaching Award—make your journey through the story of mathematics all the more riveting and exciting.

Mathematics has exhibited an inexhaustible power to illuminate aspects of the universe that have been cloaked in mystery. In charting the storied history of its evolution, The Queen of the Sciences not only illustrates how these mysteries were revealed but exposes, with a wealth of insight, the enormous efforts that went into deciphering our natural world.

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24 lectures
 |  Average 30 minutes each
  • 1
    What Is Mathematics?
    You explore the peculiar nature of mathematics. Why is it that abstractions that arose in one context can lead to unexpected insights in another? This lecture closes with a look at the major conceptual advances that are the focus of this course. x
  • 2
    Babylonian and Egyptian Mathematics
    Egyptian and Mesopotamian mathematics were well developed by the time of the earliest records from the 2nd millennium B.C. Both knew how to find areas and volumes. The Babylonians solved quadratic equations using geometric methods and knew the Pythagorean theorem. x
  • 3
    Greek Mathematics—Thales to Euclid
    This lecture surveys more than 300 years of Greek mathematics, from Thales and Pythagoras to Euclid. Euclid's Elements covers much of the mathematical knowledge of the time and is considered the most important book of mathematics ever written. x
  • 4
    Greek Mathematics—Archimedes to Hypatia
    Foremost among Greek mathematicians was Archimedes, who developed methods equivalent to the modern technique of integration. Hypatia was the first woman known to have made important contributions to mathematics and was one of the last scholars of the famous Museion at Alexandria. x
  • 5
    Astronomy and the Origins of Trigonometry
    The origins of trigonometry lie in astronomy, especially in finding the length of the chord that connects the endpoints of an arc of a circle. Hipparchus discovered a solution to this problem, that was later refined by Ptolemy who authored the great astronomical work the Almagest. x
  • 6
    Indian Mathematics—Trigonometry Blossoms
    You journey through the Gupta Empire and the great period of Indian mathematics that lasted from A.D. 320 to 1200. Along the way, you explore the significant advances that occurred in trigonometry and other mathematical fields. x
  • 7
    Chinese Mathematics—Advances in Computation
    At least as early as the 3rd century B.C., Chinese civil servants had to solve problems in surveying and collecting taxes. x
  • 8
    Islamic Mathematics—The Creation of Algebra
    Algebra was perfected here in the 9th century by the great mathematician Abu Jafar al-Kwarizmi, whose name would become the word "algorithm." x
  • 9
    Italian Algebraists Solve the Cubic
    Mathematics from the Islamic world gradually spread into Europe in the 13th century, starting with Leonardo of Pisa, also known as Fibonacci. Italian mathematicians began to make original contributions in the 16th century when they discovered how to solve the general cubic and quartic equations. x
  • 10
    Napier and the Natural Logarithm
    Working at the turn of the 17th century, John Napier found a way to facilitate calculation for astronomers by inventing logarithms. He also discovered the number now designated by the letter e. x
  • 11
    Galileo and the Mathematics of Motion
    In the early 17th century, Galileo Galilei made important innovations in the study of motion, establishing the pattern of relying on mathematical models to explore physical phenomena. René Descartes and Christiaan Huygens would build directly on his insights. x
  • 12
    Fermat, Descartes, and Analytic Geometry
    A lawyer for whom mathematics was an avocation, Pierre de Fermat was instrumental in the origins of calculus. In 1637, both Fermat and René Descartes published explanations of analytic geometry. x
  • 13
    Newton—Modeling the Universe
    Isaac Newton famously said, "If I have seen further, it is by standing on the shoulders of giants." You learn who those giants were and explore Newton's invention of calculus to explain the motions of the heavens in Principia Mathematica, published in 1687. x
  • 14
    Leibniz and the Emergence of Calculus
    Independently of Newton, Gottfried Wilhelm Leibniz discovered the techniques of calculus in the 1670s, developing the notational system still used today. x
  • 15
    Euler—Calculus Proves Its Promise
    Leonard Euler dominated 18th-century mathematics so thoroughly that his contemporaries believed he had solved all important problems. x
  • 16
    Geometry—From Alhambra to Escher
    You look at the influence of geometry on art, exploring the intriguing types of symmetry in Moorish tiling patterns. You also examine the geometrical experiments of M. C. Escher and August Möbius. x
  • 17
    Gauss—Invention of Differential Geometry
    You explore Carl Friedrich Gauss and his interest in geometry on various kinds of surfaces, including his work on the parallel postulate, which laid the foundations for non-Euclidean geometry. x
  • 18
    Algebra Becomes the Science of Symmetry
    Algebra underwent a fundamental change in the 19th century, becoming a tool for studying transformations. One of the most tragic stories in mathematics involves Evariste Galois, who invented a set of transformations before dying at age 20 in a duel. x
  • 19
    Modern Analysis—Fourier to Carleson
    By 1800, calculus was well established as a powerful tool for solving practical problems, but its logical underpinnings were shaky. You explore the creative mathematics that addressed this problem in work from Joseph Fourier in the 19th century to Lennart Carleson in the 20th. x
  • 20
    Riemann Sets New Directions for Analysis
    Bernhard Riemann left two famous legacies: the Riemann hypothesis, which deals with the distribution of prime numbers and is the most important open problem in mathematics today, and Riemann's new system of geometry, which Einstein used to develop his general theory of relativity. x
  • 21
    Sylvester and Ramanujan—Different Worlds
    This lecture explores the contrasting careers of James Joseph Sylvester, who was instrumental in developing an American mathematical tradition, and Srinivasa Ramanujan, a poor college dropout from India who produced a rich range of new mathematics during his short life. x
  • 22
    Fermat's Last Theorem—The Final Triumph
    Pierre de Fermat's enigmatic note regarding a proof that he didn't have space to write down sparked the most celebrated search in mathematics, lasting more than 350 years. This lecture follows the route to a proof, finally achieved in the 1990s. x
  • 23
    Mathematics—The Ultimate Physical Reality
    Mathematics is the key to realms outside our intuition. You begin with Maxwell's equations and continue through general relativity, quantum mechanics, and string theory to see how mathematics enables us to work with physical realities for which our experience fails us. x
  • 24
    Problems and Prospects for the 21st Century
    This last lecture introduces some of the most promising and important questions in the field and examines mathematical challenges from other disciplines, especially genetics. x

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  • 184-page printed course guidebook
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Your professor

David M. Bressoud

About Your Professor

David M. Bressoud, Ph.D.
Macalester College
Dr. David M. Bressoud is the DeWitt Wallace Professor of Mathematics in the Department of Mathematics and Computer Science at Macalester College. He earned his bachelor's degree in Mathematics from Swarthmore College and his master's degree and Ph.D. in Mathematics from Temple University. Professor Bressoud is experienced in teaching mathematics to students of all levels. As a Peace Corps volunteer before earning his Ph.D.,...
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Queen of the Sciences: A History of Mathematics is rated 4.6 out of 5 by 57.
Rated 5 out of 5 by from Wonderful course This is an excellent history of mathematics with a wonderful professor who clearly knows his subject. Professor Bressoud is a passionate teacher and makes this course interesting and inspiring. Recommended to those with some background in mathematics.
Date published: 2010-01-28
Rated 4 out of 5 by from More Math Than The Title Implies No doubt about it, this is a history course for mathematicians. Prof Bressoud seems to cover all the major mathematicians from the ancient times to modern day. Or if not, he sure covers an awful lot of them. One possible problem for you - and certainly not the professor's fault - is that you can't cover what a mathematician contributed without at least broaching the mathematics itself. This course goes through algebra, geometry, trig, calculus, number theory and other topics in math, some of which I'd never heard of. Thus, the more math you know, the more you'll likely get out of this course. If you're not a math whiz, you will find yourself straining to understand what's being presented. This is no way makes what the professor is offering useless to you. Just get the buzz-words (better, write them down and at least your impression of what they mean) and you can still see how all this weaves together as time marches on. My only criticism of this course - and the reason for four stars instead of five - is it could have been made much more enjoyable and entertaining with more graphics and especially animated graphics. There are places that fairly scream for more of these things. Also, Prof Bressoud, although clearly competent and knowledgable, is not as lively a speaker as you may be used to seeing from TTC. In short, if your knowledge of math is above average and you're interested in the players that brought the discipline this far, this is an excellent course.
Date published: 2009-11-06
Rated 4 out of 5 by from good content but too superficial as Professor Bressound said in the first lecture, his lectures would be somewhat superficial since the course time couldn't be long. for me, the only foible of this course is its superficial description of those mathematicians. when I was already fascinated by his talk of Thales, he jumped off to the next mathematician. when I imagined myself entering into a competition among Italian mathematicians, the lecture was ended and the next lecture he talked about something totally different. this course is only 12 hours long, too short for a history of mathematics. I wonder if editors of teaching company look down upon history of mathematics so they dare to give so little time for this course. In "art across the ages" there are 48 lectures, twice the time of this course,while math is much more complicated than art. this course should be at least 96 lectures long in order for us to catch the whole scene of history of mathematics and fully develope the ability of appreciation for great mathematical ideas. I am looking forward for the second edition, I hope it will be much better.
Date published: 2009-09-14
Rated 4 out of 5 by from good history, but heady towards the end The lectures up to the 17th century were insightful as to how math developed in the context of the science of the times. But after the 17th century, the topics were increasingly remote to those of us whose math background ends with calculus. More instructive visual aids would have been helpful in understanding these unfamiliar areas of math.
Date published: 2009-06-15
Rated 5 out of 5 by from This course is for everyone! I like numbers but my husband is the higher mathematics person so I ordered this course for him. I figured I would watch the first lectures then tell him it was over my head so I could work on my hobby while he was engaged BUT I was hooked from lecture 2. The graphics are really good and I NOW understand things I had learned by rote but never grasped. It is never too late to learn. The professor is shown PIP with some of the graphics and that really helps in understanding. It is nice to be able to "pause" the lecture to either think about what he is saying, read the guidebook, or discuss the information. I have enjoyed this course and so has my husband. That must mean it is for everyone!
Date published: 2009-04-20
Rated 5 out of 5 by from Lovely course in an often neglected field This is an excellent overview of the history of mathematics. The professor is extremely adept at presenting mathematical concepts to a non technical audience. My only complaint about the course, which is probably unique to mathematicians or others with a strong background in math, is that he often whizzed over deep theories with little or no explanation, referring to a theory or theorem by a name or a brief phrase of description. All of my degrees are in the field of algebra, so there were a couple of topics where I didn't have much of an idea of what he was pointing to. On the other hand, this was a good set of pointers to other areas of math that I should dig into! A common failing of graduate programs in mathematics is that there is little attention paid to the history of math. For that reason, this course was like a dish of rich chocolate bonbons! For those with less math background, this course would be a wonderful way of gaining an appreciation of what a rich history mathematics has had.
Date published: 2009-02-04
Rated 5 out of 5 by from Great lecture series I loved this lecture series. The historical details, connections between maths ideas of different cultures and the connections to science and Physics were facinating. Professor Bressoud has definetly increased my interest and motivation to learn more. A couple of minor improvements can be made: a few more graphics to replace the finger drawing of axes and concepts in the air and some more details on some of the maths, .e.g, how primes are used encrypt data on the internet. Overall, a fantastic series i would recommend to all.
Date published: 2009-02-03
Rated 5 out of 5 by from A Portion of Mathematical History I found the lecture series to be as comprehensive as one can expect on a subject four thousand years in the making and into condensed to twelve hours of lectures. Great for anyone interested in the time line of mathematics.
Date published: 2009-01-07
Rated 3 out of 5 by from Needs to Be Longer and More In-Depth I agree 100% with the review by "BGZ1." This topic and the lecturer, Bressoud, are excellent, but the coverage of the mathematics itself is too superficial to really allow a grasp of the genesis and implications of the mathematical developments, and the resulting historical narrative is fragmentary rather than coherent and flowing. Even for someone who is very interested in this topic (like me), not much of the content winds up sticking in the mind, which is a shame. And I agree that the solution is to make this course twice as long. Once again, this course illustrates that TTC shouldn't be afraid to incorporate real math into its math courses. By instead trying to appeal to the lowest common denominator among its audience, no member of the audience is actually well served. I can recommend the course only to diehard math enthusiasts, since they will best be able to overlook the limitations of the course and still extract value from it. Others may want to pass.
Date published: 2008-12-21
Rated 4 out of 5 by from Good history; math lacking This course deserved twice as much time as the professor was given. As it is, one gets a whirlwind tour of the history part, but without nearly enough of the actual mathematics to make it satisfying. Also because of time, I felt we were skimming from one topic to another, rather than being immersed in an ongoing narrative. A great topic and a very good professor. I hope the TC will consider a much broader and deeper edition.
Date published: 2008-12-19
Rated 5 out of 5 by from These courses provide me an opportunity to learn and explore areas thtat I have not had the opportunity to study. How rewarding!
Date published: 2008-10-17
Rated 5 out of 5 by from Congratulations on tracing the decimal system from China, India, & the Islamic world to Thomas Jefferson. To finalize this history, I would accept a "Dear Sir" letter from the professor inquiring about the theory of infinite decimals as real numbers.
Date published: 2008-10-17
Rated 5 out of 5 by from I have all your math courses. Totall, these courses weave common concepts in math under different contexts and viewpoints presented. This enriches the learning experience.
Date published: 2008-10-17
Rated 5 out of 5 by from Instructor explained clearly math that was well above my head. Good job!
Date published: 2008-10-17
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