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Queen of the Sciences: A History of Mathematics

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Queen of the Sciences: A History of Mathematics

Course No. 1434
Professor David M. Bressoud, Ph.D.
Macalester College
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4.6 out of 5
40 Reviews
85% of reviewers would recommend this series
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
 |  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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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 40.
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
Rated 5 out of 5 by from QUEEN OF SCIENCES: MATHEMATICSI I've just finished watching this coarse for the second time and I hope to watch it again. I'm 79 years old barely graduated college in 1960, but I've always had a fascination with mathematics and mathematicians. The lecturer was outstanding, he brought the subject to life for me.
Date published: 2017-04-23
Rated 4 out of 5 by from An adventure through Time ... I completed 'Queen of the Sciences: A History of Mathematics' a few months ago and decided to come back and share some thoughts on this lecture series. I work in the biological sciences and have not performed Math since college. However, this is NOT a mathematics course, it is a HISTORY course. A few of the previous reviewers were critical of the professor for not "teaching" a mathematics concept very well, which seems a bit unfair given that this course is about the 'history of math' and not a course on how to perform mathematical functions. Fortunately for me, I am a lover of history. I also love the sciences, so this course seemed a natural fit for me and I was not disappointed. I was mesmerized by this course and must confess that I really enjoyed it. Anyone who shares a love of history and of the sciences should find that they too, would enjoy this lectures series. The structure of this course is essentially chronological, starting with the earliest math (even speculation of pre-history math) up to the present day. The material covered near the end of the course was mostly over my head but I was still able to grasp the general concepts that the professor was focusing on. As to the professor ... Professor Bressoud has a clear and distinct manner of speaking. His pronunciation and enunciation represent a level of articulation that is second to none. You will have no trouble understanding the good professor. However, his style of speaking is also adynamic. While it is evident from his speech that he is passionate about the subject, that passion doesn't translate into the camera as well as some of his colleagues at The Great Courses. He does not exude his enthusiasm quite as visibly as those other TGC professors do. This is not a criticism, just an observation. The lectures are all organized well and build on one another, as might be expected. For me, these lectures were an Adventure through Time. I continued to marvel at the genius of mankind and what we have achieved with our brains. Professor Bressoud explains the driving concepts that motivated great thinkers to expand our understanding and use of mathematics, dividing these concepts into several broad areas: civil administration (need for counting and maintaining inventory and taxes), Astronomy (need to understand trigonometric functions), etc. In addition to explaining the driving forces that advanced the devleopment of mathematics, he also enlightens us with historical examples of how pre-existing math furthered our understanding of the sciences (physics, economics, etc). The course will start with simple arithmetic and follow the development of geometry, algebra, trigonometry and the calculus. You will learn about number systems and number theory and how math traveled from one part of the globe to another. It really is an adventure. You do not need to be a mathematician to appreciate or to follow these lectures. You will learn about many of the great Mathematicians of the past along with some interesting and entertaining anecdotal stories. Just sit back and enjoy the adventure. This will be one of those courses that I revisit, at some point in the future. It is worth a second viewing. Cheers.
Date published: 2017-04-18
Rated 3 out of 5 by from Good, but could be much better I regret that I can only marginally recommend this course. Positive aspects. The course is well thought out and each lecture makes perfect sense. The prof obviously spent lots of time preparing the syllabus. He is well spoken and and his presentation style for purely descriptive material keeps one’s interest. Negative aspects. The prof really doesn’t know how to teach anything other than descriptive type historic material. When it comes to teaching conceptual topics such as Napier’s development of logarithms all he does his waves his hands around forming imaginary lines that only he can see. His explanation of Galileo’s work is even worse. When he’s done with all that waving he puts up simple graphic of what he’s allegedly just explained. That is NOT teaching. It’s just blabbing away. He doesn’t seem to have heard of actual demonstrations, computer graphics or even a blackboard! Even with negative comments I still (barely) recommend this course. I did learn about the history of math, but it could have been so much better.
Date published: 2017-03-12
Rated 5 out of 5 by from Great Excellent review with many little known antic dotes of math history. It is obvious the professor loves his work and knows how to share his enthusiasm.Loved the course.
Date published: 2017-03-03
Rated 5 out of 5 by from No Proof Required Having taken Maths, Physics, and Chemistry in the UK equivalent of Years 11/12, I went to university to do the same three at Degree level. In the first year I realised that maths at that level was of no practical use to anybody except mathematicians ! Yet, I had always been fascinated by the development (history) of the science. Hence I purchased this course. Prof Bressoud speaks clearly and with the measured delivery one would expect of a Mathematician. The historical parts were excellent, but, I have to say, I skipped over the laborious proofs. Not, to me, in the least bit interesting...
Date published: 2016-08-22
Rated 4 out of 5 by from Excellent except for "X" Overall, the depth and breadth of coverage was excellent! As a non-mathematician, getting an idea of the kinds of things which are widely unknown by the general public, that mathematicians have discovered and continue to puzzle over, is fascinating. And as someone who was introduced to mathematics through the “new” math of the 1960’s in which history was totally ignored, I am now convinced that historic background is essential. That said, and understanding that there is a limit to how much can be covered in one course, I would like to have had a little more depth in two areas. First, a more thorough explanation of “e” and the exponential function would have been welcome. In particular, a brief discussion pointing out that a relationship exists between “e”, compound interest (and Babylonian mathematics), growth and decay in the natural world, the Fibonacci series, and the golden mean, as well as the related fact, which was covered, that the exponential function is its own derivative/integral, would provide a better understanding of why “e” is so important. I would also like to have heard a concise intuitive statement explaining why the derivative is the inverse of the integral and vice/versa. Not much to ask! A major highlight which was worth the entire cost of the course, was the Arnold/Rogness video in Lecture No. 16 showing an animation of the Mobius transformations. I came away thinking (at least) that I actually understood what it was Mobius was doing. Similar animations showing the geometry of other functions, though no doubt difficult to produce, would certainly make mathematics a lot more exciting, never mind much more comprehensible. In general, additional visual aids, perhaps three times as many, would have made these lectures more enjoyable and understandable to watch. The ones that were used were exceptionally well done, particularly those in which the lecturer was visible in cameo in one corner. A picture is certainly worth a thousand words (or hand gestures) bringing me to my only major disappointment. As a retiree, while I count myself fortunate to have access to world class “experts in field” made available by the Great Courses, life being short, I also expect the "presentation" to be of highest professional quality. In fact, I expect to be entertained! Generally, Professor Goldman’s presentation was exceptionally well done except for one distracting habit, namely, moving his hands in cadence with every syllable of every word. This takes away from the instances in which he does effectively uses hand gestures for clarifying the verbal explanation, or for emphasis. That particular mannerism is certainly not restricted to this professor, but is also present in too many of the other Great Course DVDs. I sometimes wonder if it would be better to enlist the services of a professional actor to actually perform the script since the professor, after all, doesn’t have to answer any questions.
Date published: 2016-07-18
Rated 5 out of 5 by from Superb, detailed overview Though I've taken a number of higher-level mathematics courses over the decades, this Professor awes and informs me in each lesson as he reveals the intricate, and complex development of mathematics over the centuries and across the world. In each lesson, I have at least one "Wow! I didn't know that." moment which makes me want to go back and delve deeper into one area of mathematics or another. His presentation is clear, detailed, and quite easy to follow as he weaves through the simplest and most complex areas of Mathematics. I find it almost impossible to believe this one person can so well understand and understand virtually all areas of Mathematical theory.
Date published: 2015-10-24
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