Queen of the Sciences: A History of Mathematics

Course No. 1434
Professor David M. Bressoud, Ph.D.
Macalester College
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Course No. 1434
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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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  • 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 59.
Rated 5 out of 5 by from Enjoyable! I have a master's degree in mathematics, and I found this one very enjoyable and informative. I recommend it.
Date published: 2020-07-26
Rated 5 out of 5 by from Enjoyable But A Bit Deep in Some Areas This was an interesting course on the history of math. At times it was a bit deep in some areas for me, but still enjoyable. One of the interesting themes was that some math concepts have been around for 1,000 years or more, but were unknown to many people until the Europeans started to name them and bring them to light because of the way that they communicated. An example is Pascal's Triangle which was known years before Pascal's time but apparently never explored outside of the academic world. Another example is that in the late 5th century Zu Chongzhi was working with 355/113 which was the approximation to Pi, but he had no knowlegde that Pi was being worked with in other cultures around the known world. This course also gives various problems and their solutions to an assortment of archic but still used today concepts. Overall, this course would be useful to anyone who is working on a math degree or working in a math related field and who wants more insight as to why concepts developed in the way that they did.
Date published: 2020-01-11
Rated 5 out of 5 by from Excellent and Informative First, full disclosure: I was trained as a mathematician and knew that Prof. Bressoud would do an excellent job, having read one of his books. You may think it's somewhat redundant for a mathematician to be watching a lecture series like this, but the truth is that only a small part of the information in this series is included in contemporary mathematical training, even at the advanced level. There just doesn't seem to be time for it. So I've been learning things I never knew in just about every one of the 12 lectures I've seen so far. One example would be the fact that the Pythagorean Theorem was known long before Pythagoras. Same with Pascal's Triangle. Professor Bressoud is very thorough, setting the record straight on who did what and when, including saying in some cases that we just can't be sure. In this morning's lecture on Fermat and Descartes, I learned how Pierre de Fermat got the "de" in his name, something that had never occurred to me. I think it's important to point out that Bressoud does an excellent job of explaining mathematical concepts very clearly, in a way that should be fully accessible to a non-mathematician, without any sense of dumbing down. Not an easy thing to do. This lecture series is enough of a treat that I only allow myself to view one lecture per day. I can tell by the music, the set, and the aspect ratio that this series has been around for some time. I wonder how I've managed to miss it up to now. Obviously, highly recommended.
Date published: 2019-11-10
Rated 5 out of 5 by from Excellent course Prof. Bressoud (pronounced "BRESS OO") has a pleasant, amiable presentation style and he comprehensively covers the growth of mathematics from ancient times on. The course dwells more on the many historical developments from diverse civilizations than it does on mathematical analyses, so this is not a math course. Where math formulas or equations are discussed, Prof. Bressoud presents them in straightforward, palatable examples. We greatly enjoyed learning how the field of mathematics developed, layer by layer, over the centuries with contributions by a large international cast.
Date published: 2019-10-30
Rated 5 out of 5 by from Excellent lecturer Dr. David Bressoud is about a doubt one of the finest lecturers I have ever encountered.
Date published: 2019-10-18
Rated 5 out of 5 by from Excellent Global Overview I've always had an interest in the history of science and mathematics. This courses manages to give a complete historical overview of the development of mathematics since antiquity. I recommend it to anyone with similar interests or curiosity.
Date published: 2019-09-18
Rated 5 out of 5 by from Excellent Lecturer I especially appreciate the focused attention given to each major center of mathematical thought and development in connection with astronomy (and astrology), civil administration, and navigation and surveying. You can get a real sense of the natural transition in the minds of mathematicians from practicalities to theoretical structure in Babylon and Egypt, Greece, India and China, and in Europe. The more math you know, the greater is your appreciation of these descriptions of the pioneers of modern thought. I enjoy the lecture style of Dr. Bressoud as he moves smoothly from topic to topic in a very coherent and lively way. I like to take notes during lectures and review them, taking a few minutes, right after the lecture to appreciate better the structure of the presentation. Dr. Bressoud always leaves a "teaser" to draw you into the next lecture.
Date published: 2019-09-16
Rated 5 out of 5 by from wonderful background knowledge. I have always been interested in math, and this course brings the history of , and the development of many of the topics I have studied. It is a great course (no pun intended).
Date published: 2019-05-05
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