Queen of the Sciences: A History of Mathematics

Course No. 1434
Professor David M. Bressoud, Ph.D.
Macalester College
Share This Course
4.6 out of 5
48 Reviews
87% of reviewers would recommend this product
Course No. 1434
Video Streaming Included Free

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.

Hide Full Description
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

Lecture Titles

Clone Content from Your Professor tab

What's Included

What Does Each Format Include?

Video DVD
Video Download Includes:
  • Download 24 video lectures to your computer or mobile app
  • Downloadable PDF of the course guidebook
  • FREE video streaming of the course from our website and mobile apps
Video DVD
DVD Includes:
  • 24 lectures on 4 DVDs
  • 184-page printed course guidebook
  • Downloadable PDF of the course guidebook
  • FREE video streaming of the course from our website and mobile apps

What Does The Course Guidebook Include?

Video DVD
Course Guidebook Details:
  • 184-page printed course guidebook
  • Suggested readings
  • Questions to consider
  • Timeline

Enjoy This Course On-the-Go with Our Mobile Apps!*

  • App store App store iPhone + iPad
  • Google Play Google Play Android Devices
  • Kindle Fire Kindle Fire Kindle Fire Tablet + Firephone
*Courses can be streamed from anywhere you have an internet connection. Standard carrier data rates may apply in areas that do not have wifi connections pursuant to your carrier contract.

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.,...
Learn More About This Professor
Also By This Professor


Queen of the Sciences: A History of Mathematics is rated 4.6 out of 5 by 48.
Rated 5 out of 5 by from Long Live the Queen! Professor David M. Bressoud’s lectures are a remarkable blend of fascinating history, close reasoning, and infectious passion for his subject matter. I wish I had had a teacher as engaging and clear as he is to present mathematics to me five and six decades ago. Though I did study mathematics back then, up through my second year of university, it has turned out that I did not have much practical need for algebra, trigonometry, calculus, so-called “imaginary” numbers, etc., during my subsequent working life. It feels great to be reminded of how aesthetically and philosophically pleasing those subjects can be. Do not expect “Queen of the Sciences: A History of Mathematics” to be either a how-to or a refresher course. As its title implies, the focus is on history. While sharing biographical information and cultural context about key thinkers and their discoveries, in roughly chronological order through the past 4000 years, it is true that Dr. Bressoud does also demonstrate some illustrative problems and solutions, mostly by way of explaining what kinds of things those thinkers were motivated to study, either because of their individual curiosity or because of societal spurs during particular eras. Be advised, however, that unless you have already got a familiarity with higher math, or you have a prodigy’s intuitive grasp of things mathematical, then you will need to consult texts or take other courses in order to understand more than just the gist of those illustrative examples. Of course, the biographies and general history discussed do have a certain stand-alone worth. The approach here reminds me of several presentations I’ve enjoyed on the history of music, wherein a presenter might share biographical and anecdotal information about someone like Beethoven, say, and then go on to give a revealing analysis of some sampling of Beethoven’s accomplishments, perhaps the form and themes of the First Movement of the Fifth Symphony—not a thorough review of everything Beethoven achieved and not a digression from the history of music to thorough training in music. Dr. Bressoud’s hand movements to illustrate such things as how the curve of a particular function might appear on a graph are quite good, better than those of most math professors I have known, but I must cite in reviewing this course that hand movements are too often all that is shown. I do not blame the professor for this shortcoming. It seems to me to be more a weakness in the overall staging and/or the videography for the course. Sometimes, Dr. Bressoud does appear in an inset while a graph or other display is simultaneously shown—too infrequently, in my opinion. It is said of Srinivasa Ramanujan, the Indian mathematical genius (1887–1920), that he regarded numbers as his personal friends. I have the same impression of Dr. Bressoud, am grateful for his Great Course, and highly recommend it.
Date published: 2018-09-20
Rated 4 out of 5 by from More graphics The courses need more graphics in the form of documentaries. A professor talking with no graphics is very difficult to follow.
Date published: 2018-07-29
Rated 5 out of 5 by from This course besides the excellent presentation and explanation provided background on the names I saw when studying Mathematics in undergraduate and graduate school. I also especially liked the Mobius transformation with ants crawling on it. (Picture in background.) I have not seen this since 1971. Yes I am really old, as my grand daughter informs me.
Date published: 2018-03-19
Rated 5 out of 5 by from History of Math; Not Math As someone who has a sound background in math, but is not a mathematician, I was really interested in this course. And was not disappointed. Professor Bressoud presents a fine history of math, without delving deeply into much of the math behind the mathematical concepts presented. Now this is a fine needle to thread and I think that Dr. Bressoud does so admirably. It is a hard task to not exclude those who are interested in the history (but not math itself) and keep those interested, who wish for rigorous proofs to be presented. I count my self as one of those who would have liked a bit more math, but I also acknowledge that the course would need another 12 lectures or so in order to accommodate more math, as I would not want any of the history, background or biographical material of the named mathematicians to be eliminated. And as I found this background material fascinating, clearly the course meets (for me) its goal. (Not to mention that rigor in some areas would have left me behind) There were a few times that I wished for more graphics, but the presentation of Escher’s Circle Limit III was outstanding. Other highpoints were the lecture on Euler, the all too human battle between Newton and Leibniz, the first half-dozen lectures setting the background, the individual mathematicians challenging each other during the Italian Renaissance and the weaving of Fermat’s Last Theorem into quite a bit of the last half of the course. I thought that Professor Bressoud came across as one who really knew and loved his material, but who was not particularly dynamic in his presentation. And he could have done a better job in lecture 10, describing natural logarithms and their importance. Otherwise, first rate.
Date published: 2018-03-09
Rated 5 out of 5 by from Interesting History This course was exactly what I wanted. I have often wondered, "Who was the first person that had a problem that required The Calculus to solve it"? This is a chronological journey of mathematical discoveries and the genius' behind them. I found Professor Bressoud to engaging, articulate and well prepared. At times I did wish that he would delve a little more into the math behind the topic, but that was not his intention. To supplement this course, I purchased "Great Thinkers, Great Theorems" by Professor Dunham. These two sets together answered all my questions admirably.
Date published: 2018-01-30
Rated 5 out of 5 by from Uniquely informative and engaging Professor Bressoud understands mathematics at a deep level that is unusual even for most professional mathematicians; he is particularly well versed in the relationships of mathematics to the empirical sciences. His account of the origins and history of trigonometry was a revelation! His presentations require more than passive listening; the learner must actively participate to follow the ideas and explanations. But this is an asset, not a liability!
Date published: 2017-12-04
Rated 4 out of 5 by from Good history and somes gems Prof. Bressoud promises right at the start to fill in what I always felt was missing from traditional math teaching : to examine the motivation leading to mathematical ideas and the difficulties encountered as the ideas were developed. A tall order which about half filled most of my expectations. The value of this course is that it provides the big picture, which is missing from the traditional teaching of mathematics – always segmented in unconnected blocks without any attempt to make the connection. This is not a maths course. You will hear about power series but just enough to understand why maths in antiquity went beyond calculating areas and volumes and why anyone would have bothered to work on extra dimensions to address astronomy observations. Prof. Bressoud reviews some of the big ideas from 4 millennia of mathematics history, back to Babylonia. The course is more about the history of maths and the significant milestones. Useful for someone who is already familiar with the concepts, the difference a nerd who just does the equation and absolutely no curiosity and someone who wants to learn more. You will learn how the heritage was preserved by successive transmission to other civilizations as the precious works were abandoned in their birthplace and were translated and amplified elsewhere. Why only a 4-star rating ? Mostly because of incomplete expositions and also because of the monumental works offered by The Great Courses, namely: - Grossman’s Thermodynamics - Wolfson’s Physics in Our Universe - Devaney’s Differential Equations – Visual Method These courses set the bar really high and the highest rating is limited to 5 stars. So, right off the starting line Prof. Bressoud’s work is not of the same scale as my three gold standards. Tough to be a star in a roomful of aces. Just do not let my less-than-perfect rating of this course discourage you. This course is beyond “good”, there are gems in there despite some less than lively delivery of the lectures. Lecture 16 is one such gem. A graphic illustration of the concept of transformation which I never saw anywhere even though I had to use them. Another gem is Lecture 18 which establishes transformations as the key to finding exact roots of higher-degree polynomials and provided my first exposure to the importance of symmetry in mathematics. Lecture 23 adds some context information to Maxwell’s equations of electromagnetism even though I did use them extensively to study propagation of microwave frequencies in rectangular waveguides. The graphical explanation of the retrograde path of Mars is very good. These gems and the frequent references to Escher Circle 3 Limit graph make me forgive the shortcoming of Lecture 10 which deals with logarithms. The explanation of how Napier calculated logarithms to simplify calculations on astronomic data. This is done without any graphics and is particularly dry. Sahara high noon dry. The arguments make sense but the whole demonstration remains obscure. Not cool. Words without pictures can only carry so far – not far enough in this case. I had been wondering how logarithms were calculated way before HP’s extraordinary HP 35 pocket calculator (1974) – I would have really appreciated a more detailed demonstration on how Napier only needed to perform about 200 calculations which were the basis for the extended table. Similarly, Lecture 19 covers the Fourier series but in a superficial way. Being myself familiar with this concept, I could almost hear newcomers wondering in the back of the classroom “so what ?” as this lecture begs for the explanation of how in infinite series actually makes calculations simpler. This lecture screams for a bit less math theory and at least a basic practical use of this powerful tool. Another flaw is in Lecture 23 – I cannot understand why the 4 beautiful equations were not shown. And for all the talk about symmetry, the absence of Emmy Noether and her deeply profound linking symmetries and conservation laws of physics is inexplicable. Lecture 20 on the zeta function and Riemann’s hypothesis was especially dry, catering more to a mathematician than to an engineer. Differential geometry is mentioned but also as a dry concept without any illustration. I would have gladly traded the whole lecture for a more concrete illustration of differential geometry. Overall, I did get my money’s worth with this course. I recognize that some people will find it rough sailing at times.
Date published: 2017-12-03
Rated 5 out of 5 by from very good presentation Instructor does a good job of presenting the history of mathematics. He gives a clear and organized presentation cover a fast amount of information.
Date published: 2017-05-10
  • y_2018, m_12, d_17, h_19
  • bvseo_bulk, prod_bvrr, vn_bulk_2.0.9
  • cp_1, bvpage1
  • co_hasreviews, tv_4, tr_44
  • loc_en_US, sid_1434, prod, sort_[SortEntry(order=SUBMISSION_TIME, direction=DESCENDING)]
  • clientName_teachco
  • bvseo_sdk, p_sdk, 3.2.0
  • CLOUD, getContent, 19.98ms

Questions & Answers

Customers Who Bought This Course Also Bought

Buy together as a Set
Choose a Set Format