## Archimedes’ Constant

The number pi (π) can be simply stated: it is the ratio of a circle’s circumference (C) to its diameter (d).

$\pi = \frac{C}{d}$

The Greek mathematician Archimedes (c. 287 BC – c. 212 BC) was the first person to come up with a computational method of calculating π. He inscribed and circumscribed polygons with the same number of sides inside and outside of a circle. The value of π is between the perimeter of the inscribed polygon and the perimeter of the circumscribed polygon as shown in the diagrams below. By increasing the number of sides of the inscribed and circumscribed polygons, the value of π can be estimated more closely. The number π is thus sometimes called Archimedes’ Constant.

Archimedes’ Constant was not called π until Welsh mathematician William Jones (1675-1749) began using it in 1706. π is the first letter of the Greek word for periphery (περιφέρεια).

The number π (3.1415926535897932384626433…) has some remarkable properties, a few of which are

• π cannot be expressed as a ratio of two integers (it is an irrational number).
• The exact decimal representation of π has an infinite number of digits.
• The decimal digits of π never exhibit a repeating pattern.
• The decimal digits of π are randomly distributed, but this has not yet been proven.
• π cannot be a solution of any equation involving only sums, products, exponents, and integers (it is a transcendental number).

It is worth noting that there are an infinite number of transcendental numbers (and, therefore, at least an infinite number of irrational numbers). But π is remarkable in that it pervades both mathematics and physics, often in ways that appears to have nothing to do with circles, spheres, or even geometry.

The value of π has now been calculated out to 100 trillion decimal places (1014) by Japanese computer scientist Emma Haruka Iwao. Like other recent attempts to calculate the most digits of π, Iwao used the Chudnovsky algorithm. Her record-breaking calculation took nearly 158 days using cloud computing between October 14, 2021 and March 21, 2o22.

Interestingly, the value of π can be calculated using a couple of simple infinite series.

The great Swiss mathematician Leonhard Euler (1707-1783) obtained the following:

$\pi = \sqrt{6\left ( \frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots \right )}$

And earlier, German mathematician Gottfried Wilhelm Leibniz (1646-1716) and Scottish mathematician David Gregory (1659-1708) independently arrived at an even simpler infinite series to generate π, though it converges so slowly that it is of little practical use.

$\pi = 4\left ( \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots \right )$

## Great Courses, Great Episodes

The Great Courses offers a number of excellent courses on DVD (also streaming and audio only). Here are my favorite episodes. (Note: This is a work in progress and more entries will be added in the future.)

Course No. 153
Einstein’s Relativity and the Quantum Revolution: Modern Physics for Non-Scientists, 2nd Edition – Richard Wolfson
Lecture 8 – Uncommon Sense—Stretching Time
“Why does the simple statement of relativity—that the laws of physics are the same for all observers in uniform motion—lead directly to absurd-seeming situations that violate our commonsense notions of space and time?”
Lecture 9 – Muons and Time-Traveling Twins
“As a dramatic example of what relativity implies, you will consider a thought experiment involving a pair of twins, one of whom goes on a journey to the stars and returns to Earth younger than her sister!”
Lecture 12 – What about E=mc2 and is Everything Relative?
“Shortly after publishing his 1905 paper on special relativity, Einstein realized that his theory required a fundamental equivalence between mass and energy, which he expressed in the equation E=mc2. Among other things, this famous formula means that the energy contained in a single raisin could power a large city for an entire day.”
Lecture 16 – Into the Heart of Matter
“With this lecture, you turn from relativity to explore the universe at the smallest scales. By the early 1900s, Ernest Rutherford and colleagues showed that atoms consist of a positively charged nucleus surrounded by negatively charged electrons whirling around it. But Rutherford’s model could not explain all the observed phenomena.”
Lecture 19 – Quantum Uncertainty—Farewell to Determinism
“Quantization places severe limits on our ability to observe nature at the atomic scale because it implies that the act of observation disturbs that which is being observed. The result is Werner Heisenberg’s famous Uncertainty Principle. What exactly does this principle say, and what are the philosophical implications?”
Lecture 21 – Quantum Weirdness and Schrödinger’s Cat
“Wave-particle duality gives rise to strange phenomena, some of which are explored in Schrödinger’s famous ‘cat in the box’ example. Philosophical debate on Schrödinger’s cat still rages.”

Course No. 158
My Favorite Universe – Neil deGrasse Tyson
Lecture 8 – In Defense of the Big Bang
“We now know without doubt how the universe began, how it evolved, and how it will end. This lecture explains and defends a “theory” far too often misunderstood.”

Course No. 415
The Will to Power: The Philosophy of Friedrich Nietzsche
Robert C. Solomon & Kathleen M. Higgins

Lecture 7 – Nietzsche and Schopenhauer on Pessimism
“Schopenhauer, the severe pessimist, is a looming presence in Nietzsche’s thought. Nietzsche felt the weight of Schopenhauer’s pessimism, and struggled to counter it by embracing “cheerfulness,” creative passion, and an aesthetic viewpoint.”
Lecture 19 – The Ranking of Values – Morality and Modernity
“Why did Nietzsche refuse to think of values as being either objective or subjective? Why did he hold that values are earthly and culture- and species-specific? Why did he argue that, in the final analysis, there are only healthy and unhealthy values, and that modern values are unhealthy?”
Lecture 22 – Resentment, Revenge, and Justice
“We continue our discussion of Nietzsche’s idea of resentment, adding to it his ideas about revenge and justice. We revisit his condemnation of asceticism, the self-denial that is often a part of extreme religious practice, in light of these new ideas.”

Course No. 443
Power over People: Classical and Modern Political Theory – Dennis Dalton
Lecture 10 – Marx’s Critique of Capitalism and the Solution of Communism
“Karl Marx’s communism provided what is probably the best known ideal society. He blamed not only private property, but the entire institution of capitalism for the inequality and injustice in society. His program has never been implemented, certainly not in the Soviet Union. Marx never advocated totalitarian or despotic rule. Although his historical determinism has been discredited, his social criticism remains relevant. The democratic dilemma boils down to this: the more liberty, the less equality; and the more equality, the less liberty.”
Special Note: I will eventually be adding more of the episodes from this excellent course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 700
How to Listen to and Understand Great Music, 3rd Edition – Robert Greenberg
Lecture 23 – Classical-era Form—Sonata Form, Part 1
“In Lectures 23 and 24 we examine sonata-allegro form, but first, we observe the life and personality of the extraordinary Wolfgang Mozart. We discuss the many meanings and uses of the word “sonata.” The fourth movement of Mozart’s Symphony in G Minor, K. 550, is analyzed and discussed in depth as an example.”
Special Note: I will eventually be adding more of the episodes from this excellent course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 730
Symphonies of Beethoven – Robert Greenberg
Lecture 11 – Symphony No. 3—The “New Path”—Heroism and Self-Expression, III
“Lectures 9 through 12 focus on Symphony No. 3, the Eroica Symphony. This key work in Beethoven’s compositional revolution resulted from his crisis of going deaf. Beethoven’s struggle with his disability raised him to a new level of creativity. Symphony No. 3 parallels his heroic battle with and ultimate triumph over adversity. The symphony’s debt to Napoleon is discussed before an analysis.”
Lecture 13 – Symphony No. 4—Consolidation of the New Aesthetic, I
“Lectures 13 through 16 examine Symphony No. 4 in historical context and in its relationship to opera buffa. Symphony No. 4 is the most infrequently heard of his symphonies. We see how it represents a return to a Classical structure. Its framework is filled with iconoclastic rhythms, harmonies, and characteristic motivic developments that mark it as a product of Beethoven’s post-Eroica period.”
Lecture 23 – Symphony No. 7—The Symphony as Dance, I
Lecture 24 – Symphony No. 7—The Symphony as Dance, II
“Lectures 23 and 24 discuss Beethoven’s Symphony No. 7 with references to the historical and personal events surrounding its composition. The essence of the symphony is seen to be the power of rhythm, and originality is seen to be an important artistic goal for Beethoven.”
Lecture 31 – Symphony No. 9—The Symphony as the World, IV
“The last five lectures are devoted to Symphony No. 9, the most influential Western musical composition of the 19th century and the most influential symphony ever written. We see how this work obliterated distinctions between the instrumental symphony and dramatic vocal works such as opera. Also discussed are Beethoven’s fall from public favor in 1815, his disastrous relationship with his nephew Karl, his artistic rebirth around 1820, his late compositions, and his death in 1827.”

Course No. 753
Great Masters: Tchaikovsky-His Life and Music – Robert Greenberg
Lecture 1 – Introduction and Early Life
“Tchaikovsky was an extremely sensitive child, obsessive about music and his mother. His private life was reflected to a rare degree in his music. His mother’s death when he was 14 years old was a shattering experience for him—one that found poignant expression in his music.”
Lecture 6 – My Great Friend
“With the generous financial support of Nadezhda von Meck, Tchaikovsky lived abroad, and in 1878 resigned from the Moscow Conservatory to compose full time. His Fourth Symphony was premiered in Moscow and was quickly followed by the brilliant Violin Concerto in D Major, which became a pillar of the repertoire within a few years.”

Course No. 754
Great Masters: Stravinsky-His Life and Music – Robert Greenberg
Lecture 2 – From Student to Professional
“Rimsky-Korsakov was so impressed with Stravinsky’s Piano Sonata in F♯ minor (1904) he agreed to take Stravinsky as a private student. In 1909, Stravinsky met the impresario Serge Diaghilev, who commissioned Stravinsky to write a ballet on the folk tale The Firebird, which was followed by the ballet Petrushka, a great success. Stravinsky’s next score, The Rite of Spring, would become arguably the most influential work of its time.”

Course No. 756
Great Masters: Mahler-His Life and Music – Robert Greenberg
Lecture 7 – Symphony No. 6, and Das Lied von der Erde
“Three events shattered the Mahlers’ lives in 1907: his resignation from the Royal Vienna Opera, the death of their elder daughter, and the diagnosis of his heart disease. In 1908, Mahler threw himself into composing Das Lied von der Erde as an attempt to find solace from the grief of his daughter’s death. The work is a symphonic song cycle about loss, grief, memory, disintegration, and transfiguration.”

Course No. 758
Great Masters: Liszt-His Life and Music – Robert Greenberg
Lecture 2 – A Born Pianist
“Liszt was surrounded by music from infancy and began to reveal his musical gifts at about age five. He stunned his teachers and, at his first performance at age 11, astonished reviewers and his audience. When Liszt was 15, his father died, sending Franz into depression and apathy for three years. He was finally blasted out of his lethargy by the July Revolution of 1830.”
Lecture 7 – Rome
“By the 1850s, Liszt became the focal point of a debate concerning program music versus absolute music and expression versus structure. Twenty years before, Liszt and his fellow young Romantic musicians had a common goal: to create a new music based on individual expression. As they grew older, many became conservative, but Liszt never lost his revolutionary spirit. But brokenhearted by the death of his daughter, he turned to the Catholic Church to find solace.”

Course No. 759
Great Masters: Robert and Clara Schumann-Their Lives and Music – Robert Greenberg
“In Düsseldorf, Robert was inspired to write the Symphony No. 3 in E-flat Major, along with trios, sonatas, orchestral works, and pieces for chorus and voice and piano. Robert and Clara also met Johannes Brahms there; he became a lifelong friend and source of strength for Clara. In 1854, Robert attempted to drown himself in the Rhine and was taken to an asylum. He died there two years later. Clara managed to sustain the family through her concerts but was dealt even more pain by the early deaths of several of her children.”

Course No. 1012
Chemistry, 2nd Edition – Frank Cardulla
Lecture 5 – The SI (Metric) System of Measurement
“Next, we continue to lay a strong foundation for our understanding of chemistry by learning about one of the key tools we’ll be using: the International System of Units (SI), or the metric system. This lecture explains why this system is so useful to scientists and lays out the prefixes and units of measurement that make up the metric system.”
Lecture 10 – The Mole
“One of the most important concepts to master in an introductory chemistry course is the concept of the mole, which provides chemists with a way to ‘count’ atoms and molecules. Learn how scientists use the mole and explore the quantitative definition of this basic unit.”
Lecture 28 – The Self-Ionization of Water
“After examining how different substances may behave when dissolved in water, we learn about the self-ionization of water and use this knowledge to solve problems. The lecture ends with a brief introduction to the pH of solutions.”
Lecture 29 – Strong Acids and Bases – General Properties
“We return to the topic of pH and learn about how pH relates to two kinds of compounds: acids and bases. Through an introductory problem, we explore the relationship of various ions within these compounds.”

Course No. 1257
Mysteries of Modern Physics: Time – Sean Carroll
Lecture 10 – Playing with Entropy
“Sharpen your understanding of entropy by examining different macroscopic systems and asking, which has higher entropy and which has lower entropy? Also evaluate James Clerk Maxwell’s famous thought experiment about a demon who seemingly defies the principle that entropy always increases.”
Lecture 15 – The Perception of Time
“Turn to the way humans perceive time, which can vary greatly from clock time. In particular, focus on experiments that shed light on our time sense. For example, tests show that even though we think we perceive the present moment, we actually live 80 milliseconds in the past.”
Lecture 16 – Memory and Consciousness
“Remembering the past and projecting into the future are crucial for human consciousness, as shown by cases where these faculties are impaired. Investigate what happens in the brain when we remember, exploring different kinds of memory and the phenomena of false memories and false forgetting.”
Lecture 20 – Black Hole Entropy
“Stephen Hawking showed that black holes emit radiation and therefore have entropy. Since the entropy in the universe today is overwhelmingly in the form of black holes and there were no black holes in the early universe, entropy must have been much lower in the deep past.”
Lecture 21 – Evolution of the Universe
“Follow the history of the universe from just after the big bang to the far future, when the universe will consist of virtually empty space at maximum entropy. Learn what is well founded and what is less certain about this picture of a universe winding down.”

Course No. 1280
Physics and Our Universe: How It All Works – Richard Wolfson
Lecture 1 – The Fundamental Science

“Take a quick trip from the subatomic to the galactic realm as an introduction to physics, the science that explains physical reality at all scales. Professor Wolfson shows how physics is the fundamental science that underlies all the natural sciences. He also describes phenomena that are still beyond its explanatory power.”
Lecture 24 – The Ideal Gas
“Delve into the deep link between thermodynamics, which looks at heat on the macroscopic scale, and statistical mechanics, which views it on the molecular level. Your starting point is the ideal gas law, which approximates the behavior of many gases, showing how temperature, pressure, and volume are connected by a simple formula.”
Lecture 44 – Cracks in the Classical Picture
“Embark on the final section of the course, which covers the revolutionary theories that superseded classical physics. Why did classical physics need to be replaced? Discover that by the late 19th century, inexplicable cracks were beginning to appear in its explanatory power.”
Lecture 48 – Space-Time and Mass-Energy
“In relativity theory, contrary to popular views, reality is what’s not relative—that is, what doesn’t depend on one’s frame of reference. See how space and time constitute one such pair, merging into a four-dimensional space-time. Mass and energy similarly join, related by Einstein’s famous E = mc2.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 1360
Introduction to Astrophysics – Joshua Winn
Lecture 5 – Newton’s Hardest Problem
“Continue your exploration of motion by discovering the law of gravity just as Newton might have—by analyzing Kepler’s laws with the aid of calculus (which Newton invented for the purpose). Look at a graphical method for understanding orbits, and consider the conservation laws of angular momentum and energy in light of Emmy Noether’s theory that links conservation laws and symmetry.”
Lecture 10 – Optical Telescopes
“Consider the problem of gleaning information from the severely limited number of optical photons originating from astronomical sources. Our eyes can only do it so well, and telescopes have several major advantages: increased light-gathering power, greater sensitivity of telescopic cameras and sensors such as charge-coupled devices (CCDs), and enhanced angular and spectral resolution.”
Lecture 11 – Radio and X-Ray Telescopes
“Non-visible wavelengths compose by far the largest part of the electromagnetic spectrum. Even so, many astronomers assumed there was nothing to see in these bands. The invention of radio and X-ray telescopes proved them spectacularly wrong. Examine the challenges of detecting and focusing radio and X-ray light, and the dazzling astronomical phenomena that radiate in these wavelengths.”
Lecture 12 – The Message in a Spectrum
“Starting with the spectrum of sunlight, notice that thin dark lines are present at certain wavelengths. These absorption lines reveal the composition and temperature of the Sun’s outer atmosphere, and similar lines characterize other stars. More diffuse phenomena such as nebulae produce bright emission lines against a dark spectrum. Probe the quantum and thermodynamic events implied by these clues.”
Lecture 13 – The Properties of Stars
“Take stock of the wide range of stellar luminosities, temperatures, masses, and radii using spectra and other data. In the process, construct the celebrated Hertzsprung–Russell diagram, with its main sequence of stars in the prime of life, including the Sun. Note that two out of three stars have companions. Investigate the orbital dynamics of these binary systems.”
Lecture 15 – Why Stars Shine
“Get a crash course in nuclear physics as you explore what makes stars shine. Zero in on the Sun, working out the mass it has consumed through nuclear fusion during its 4.5-billion-year history. While it’s natural to picture the Sun as a giant furnace of nuclear bombs going off non-stop, calculations show it’s more like a collection of toasters; the Sun is luminous simply because it’s so big.”
Lecture 16 – Simple Stellar Models
“Learn how stars work by delving into stellar structure, using the Sun as a model. Relying on several physical principles and sticking to order-of-magnitude calculations, determine the pressure and temperature at the center of the Sun, and the time it takes for energy generated in the interior to reach the surface, which amounts to thousands of years. Apply your conclusions to other stars.”
Lecture 17 – White Dwarfs
“Discover the fate of solar mass stars after they exhaust their nuclear fuel. The galaxies are teeming with these dim “white dwarfs” that pack the mass of the Sun into a sphere roughly the size of Earth. Venture into quantum theory to understand what keeps these exotic stars from collapsing into black holes, and learn about the Chandrasekhar limit, which determines a white dwarf’s maximum mass.”
Lecture 18 – When Stars Grow Old
“Trace stellar evolution from two points of view. First, dive into a protostar and witness events unfold as the star begins to contract and fuse hydrogen. Exhausting that, it fuses heavier elements and eventually collapses into a white dwarf—or something even denser. Next, view this story from the outside, seeing how stellar evolution looks to observers studying stars with telescopes.”
Lecture 19 – Supernovas and Neutron Stars
“Look inside a star that weighs several solar masses to chart its demise after fusing all possible nuclear fuel. Such stars end in a gigantic explosion called a supernova, blowing off outer material and producing a super-compact neutron star, a billion times denser than a white dwarf. Study the rapid spin of neutron stars and the energy they send beaming across the cosmos.”
Lecture 20 – Gravitational Waves
“Investigate the physics of gravitational waves, a phenomenon predicted by Einstein and long thought to be undetectable. It took one of the most violent events in the universe—colliding black holes—to generate gravitational waves that could be picked up by an experiment called LIGO on Earth, a billion light years away. This remarkable achievement won LIGO scientists the 2017 Nobel Prize in Physics.”

Course No. 1434
The Queen of the Sciences: A History of Mathematics – David M. Bressoud
Lecture 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.”
Lecture 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.”
Lecture 6 – Indian Mathematics – Trigonometry Blossoms
“We journey through the Gupta Empire and the great period of Indian mathematics that lasted from A.D. 320 to 1200. Along the way, we explore the significant advances that occurred in trigonometry and other mathematical fields.”
Lecture 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.”
Lecture 15 – Euler – Calculus Proves Its Promise
“Leonhard Euler dominated 18th-century mathematics so thoroughly that his contemporaries believed he had solved all important problems.”
Lecture 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. We explore the creative mathematics that addressed this problem in work from Joseph Fourier in the 19th century to Lennart Carleson in the 20th.”
Lecture 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.”
Lecture 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.”
Lecture 23 – Mathematics – The Ultimate Physical Reality
“Mathematics is the key to realms outside our intuition. We 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.”
Lecture 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.”

Course No. 1456
Discrete Mathematics – Arthur T. Benjamin
Lecture 8 – Linear Recurrences and Fibonacci Numbers
“Investigate some interesting properties of Fibonacci numbers, which are defined using the concept of linear recurrence. In the 13th century, the Italian mathematician Leonardo of Pisa, called Fibonacci, used this sequence to solve a problem of idealized reproduction in rabbits.”
Lecture 15 – Open Secrets—Public Key Cryptography
“The idea behind public key cryptography sounds impossible: The key for encoding a secret message is publicized for all to know, yet only the recipient can reverse the procedure. Learn how this approach, widely used over the Internet, relies on Euler’s theorem in number theory.”
Lecture 16 – The Birth of Graph Theory
“This lecture introduces the last major section of the course, graph theory, covering the basic definitions, notations, and theorems. The first theorem of graph theory is yet another contribution by Euler, and you see how it applies to the popular puzzle of drawing a given shape without lifting the pencil or retracing any edge.”
Lecture 18 – Social Networks and Stable Marriages
“Apply graph theory to social networks, investigating such issues as the handshake theorem, Ramsey’s theorem, and the stable marriage theorem, which proves that in any equal collection of eligible men and women, at least one pairing exists for each person so that no extramarital affairs will take place.”
Lecture 20 – Weighted Graphs and Minimum Spanning Trees
“When you call someone on a cell phone, you can think of yourself as a leaf on a giant ‘tree’—a connected graph with no cycles. Trees have a very simple yet powerful structure that make them useful for organizing all sorts of information.”
Lecture 22 – Coloring Graphs and Maps
“According to the four-color theorem, any map can be colored in such a way that no adjacent regions are assigned the same color and, at most, four colors suffice. Learn how this problem went unsolved for centuries and has only been proved recently with computer assistance.”

Course No. 1471
Great Thinkers, Great Theorems – William Dunham
Lecture 5 – Number Theory in Euclid
“In addition to being a geometer, Euclid was a pioneering number theorist, a subject he took up in books VII, VIII, and IX of the Elements. Focus on his proof that there are infinitely many prime numbers, which Professor Dunham considers one of the greatest proofs in all of mathematics.”
Lecture 6 – The Life and Work of Archimedes
“Even more distinguished than Euclid was Archimedes, whose brilliant ideas took centuries to fully absorb. Probe the life and famous death of this absent-minded thinker, who once ran unclothed through the streets, shouting ‘Eureka!’ (‘I have found it!’) on solving a problem in his bath.”
Lecture 7 – Archimedes’ Determination of Circular Area
“See Archimedes in action by following his solution to the problem of determining circular area—a question that seems trivial today but only because he solved it so simply and decisively. His unusual strategy relied on a pair of indirect proofs.”
Lecture 8 – Heron’s Formula for Triangular Area
“Heron of Alexandria (also called Hero) is known as the inventor of a proto-steam engine many centuries before the Industrial Revolution. Discover that he was also a great mathematician who devised a curious method for determining the area of a triangle from the lengths of its three sides.”
Lecture 9 – Al-Khwarizmi and Islamic Mathematics
“With the decline of classical civilization in the West, the focus of mathematical activity shifted to the Islamic world. Investigate the proofs of the mathematician whose name gives us our term ‘algorithm’: al-Khwarizmi. His great book on equation solving also led to the term ‘algebra.'”
Lecture 10 – A Horatio Algebra Story
“Visit the ruthless world of 16th-century Italian universities, where mathematicians kept their discoveries to themselves so they could win public competitions against their rivals. Meet one of the most colorful of these figures: Gerolamo Cardano, who solved several key problems. In secret, of course.”
Lecture 11 – To the Cubic and Beyond
“Trace Cardano’s path to his greatest triumph: the solution to the cubic equation, widely considered impossible at the time. His protégé, Ludovico Ferrari, then solved the quartic equation. Norwegian mathematician Niels Abel later showed that no general solutions are possible for fifth- or higher-degree equations.”
Lecture 12 – The Heroic Century
“The 17th century saw the pace of mathematical innovations accelerate, not least in the introduction of more streamlined notation. Survey the revolutionary thinkers of this period, including John Napier, Henry Briggs, René Descartes, Blaise Pascal, and Pierre de Fermat, whose famous ‘last theorem’ would not be proved until 1995.”
Lecture 13 – The Legacy of Newton
“Explore the eventful life of Isaac Newton, one of the greatest geniuses of all time. Obsessive in his search for answers to questions from optics to alchemy to theology, he made his biggest mark in mathematics and science, inventing calculus and discovering the law of universal gravitation.”
Lecture 14 – Newton’s Infinite Series
“Start with the binomial expansion, then turn to Newton’s innovation of using fractional and negative exponents to calculate roots—an example of his creative use of infinite series. Also see how infinite series allowed Newton to approximate sine values with extraordinary accuracy.”
Lecture 16 – The Legacy of Leibniz
“Probe the career of Newton’s great rival, Gottfried Wilhelm Leibniz, who came relatively late to mathematics, plunging in during a diplomatic assignment to Paris. In short order, he discovered the ‘Leibniz series’ to represent π, and within a few years he invented calculus independently of Newton.”
Lecture 17 – The Bernoullis and the Calculus Wars
“Follow the bitter dispute between Newton and Leibniz over priority in the development of calculus. Also encounter the Swiss brothers Jakob and Johann Bernoulli, enthusiastic supporters of Leibniz. Their fierce sibling rivalry extended to their competition to outdo each other in mathematical discoveries.”
Lecture 18 – Euler, the Master
“Meet history’s most prolific mathematician, Leonhard Euler, who went blind in his sixties but kept turning out brilliant papers. A sampling of his achievements: the number e, crucial in calculus; Euler’s identity, responsible for the most beautiful theorem ever; Euler’s polyhedral formula; and Euler’s path.”
Lecture 19 – Eulers Extraordinary Sum
“Euler won his spurs as a great mathematician by finding the value of a converging infinite series that had stumped the Bernoulli brothers and everyone else who tried it. Pursue Euler’s analysis through the twists and turns that led to a brilliantly simple answer.”
Lecture 20 – Euler and the Partitioning of Numbers
“Investigate Euler’s contribution to number theory by first warming up with the concept of amicable numbers—a truly rare breed of integers until Euler vastly increased the supply. Then move on to Euler’s daring proof of a partitioning property of whole numbers.”
Lecture 21 – Gauss – the Prince of Mathematicians
“Dubbed the Prince of Mathematicians by the end of his career, Carl Friedrich Gauss was already making major contributions by his teen years. Survey his many achievements in mathematics and other fields, focusing on his proof that a regular 17-sided polygon can be constructed with compass and straightedge alone.”
Lecture 22 – The 19th Century – Rigor and Liberation
“Delve into some of the important trends of 19th-century mathematics: a quest for rigor in securing the foundations of calculus; the liberation from the physical sciences, embodied by non-Euclidean geometry; and the first significant steps toward opening the field to women.”
Lecture 23 – Cantor and the Infinite
“Another turning point of 19th-century mathematics was an increasing level of abstraction, notably in the approach to the infinite taken by Georg Cantor. Explore the paradoxes of the ‘completed’ infinite, and how Cantor resolved this mystery with transfinite numbers, exemplified by the transfinite cardinal aleph-naught.”
Lecture 24 – Beyond the Infinite
“See how it’s possible to build an infinite set that’s bigger than the set of all whole numbers, which is itself infinite. Conclude the course with Cantor’s theorem that the transcendental numbers greatly outnumber the seemingly more abundant algebraic numbers—a final example of the elegance, economy, and surprise of a mathematical masterpiece.”

Course No. 1495
Introduction to Number Theory – Edward B. Burger
Lecture 12 – The RSA Encryption Scheme
“We continue our consideration of cryptography and examine how Fermat’s 350-year-old theorem about primes applies to the modern technological world, as seen in modern banking and credit card encryption.”
Lecture 22 – Writing Real Numbers as Continued Fractions
“Real numbers are often expressed as endless decimals. Here we study an algorithm for writing real numbers as an intriguing repeated fraction-within-a-fraction expansion. Along the way, we encounter new insights about the hidden structure within the real numbers.”
Lecture 24 – A Journey’s End and the Journey Ahead
“In this final lecture, we take a step back to view the entire panorama of number theory and celebrate some of the synergistic moments when seemingly unrelated ideas came together to tell a unified story of number.”

Course No. 1499
Zero to Infinity: A History of Numbers – Edward B. Burger
Lecture 2 – The Dawn of Numbers
“One of the earliest questions was “How many?” Humans have been answering this question for thousands of years—since Sumerian shepherds used pebbles to keep track of their sheep, Mesopotamian merchants kept their accounts on clay tablets, and Darius of Persia used a knotted cord as a calendar.”
Lecture 3 – Speaking the Language of Numbers
“As numbers became useful to count and record as well as calculate and predict, many societies, including the Sumerians, Egyptians, Mayans, and Chinese, invented sophisticated numeral systems; arithmetic developed. Negative numbers, Arabic numerals, multiplication, and division made number an area for abstract, imaginative study as well as for everyday use.”
Lecture 4 – The Dramatic Digits – The Power of Zero
“When calculation became more important, zero—a crucial breakthrough—was born. Unwieldy additive number systems, like Babylonian nails and dovetails, or Roman numerals, gave way to compact place-based systems. These systems, which include the modern base-10 system we use today, made modern mathematics possible.”
Lecture 6 – Nature’s Numbers – Patterns Without People
“Those who studied them found numbers captivating and soon realized that numerical structure, pattern, and beauty existed long before our ancestors named the numbers. In this lecture, our studies of pattern and structure in nature lead us to Fibonacci numbers and to connect them in turn to the golden ratio studied by the Pythagoreans centuries earlier.”
Lecture 7 – Numbers of Prime Importance
“Now we study prime numbers, the building blocks of all natural (counting) numbers larger than 1. This area of inquiry dates to ancient Greece, where, using one of the most elegant arguments in all of mathematics, Euclid proved that there are infinitely many primes. Some of the great questions about primes still remain unanswered; the study of primes is an active area of research known as analytic number theory.”
Lecture 8 – Challenging the Rationality of Numbers
“Babylonians and Egyptians used rational numbers, better known as fractions, perhaps as early as 2000 B.C. Pythagoreans believed rational and natural numbers made it possible to measure all possible lengths. When the Pythagoreans encountered lengths not measurable in this way, irrational numbers were born, and the world of number expanded.”
Lecture 9 – Walk the (Number) Line
“We have learned about natural numbers, integers, rational numbers, and irrationals. In this lecture, we’ll encounter real numbers, an extended notion of number. We’ll learn what distinguishes rational numbers within real numbers, and we’ll also prove that the endless decimal 0.9999… exactly equals 1.”
Lecture 10 – The Commonplace Chaos Among Real Numbers
“Rational and irrational numbers have a defining difference that leads us to an intuitive and correct conclusion, and to a new understanding about how common rationals and irrationals really are. Examining random base-10 real numbers introduces us to “normal” numbers and shows that “almost all” real numbers are normal and “almost all” real numbers are, in fact, irrational.”
Lecture 11 – A Beautiful Dusting of Zeroes and Twos
“In base-3, real numbers reveal an even deeper and more amazing structure, and we can detect and visualize a famous, and famously vexing, collection of real numbers—the Cantor Set first described by German mathematician Georg Cantor in 1883.”
Lecture 12 – An Intuitive Sojourn Into Arithmetic
“We begin with a historical overview of addition, subtraction, multiplication, division, and exponentiation, in the course of which we’ll prove why a negative number times a negative number equals a positive number. We’ll revisit Euclid’s Five Common Notions (having learned in Lecture 11 that one of these notions is not always true), and we’ll see what happens when we raise a number to a fractional or irrational power.”
Lecture 13 – The Story of π
“Pi is one of the most famous numbers in history. The Babylonians had approximated it by 1800 B.C., and computers have calculated it to the trillions of digits, but we’ll see that major questions about this amazing number remain unanswered.”
Lecture 14 – The Story of Euler’s e
“Compared to π, e is a newcomer, but it quickly became another important number in mathematics and science. Now known as Euler’s number, it is fundamental to understanding growth. This lecture traces the evolution of e.”
Lecture 15 – Transcendental Numbers
“π and e take us into the mysterious world of transcendental numbers, where we’ll learn the difference between algebraic numbers, known since the Babylonians, and the new—and teeming—realm of transcendentals.”
Lecture 16 – An Algebraic Approach to Numbers
“This part of the course invites us to take two views of number, the algebraic and the analytical. The algebraic perspective takes us to imaginary numbers, while the analytical perspective challenges our sense of what number even means.”
Lecture 17 – The Five Most Important Numbers
“Looking at complex numbers geometrically shows a way to connect the five most important numbers in mathematics: 0, 1, π, e, and i, through the most beautiful equation in mathematics, Euler’s identity.”
Lecture 19 – A New Breed of Numbers
“Pythagoreans found irrational numbers not only counterintuitive but threatening to their world-view. In this lecture, we’ll get acquainted with—and use—some numbers that we may find equally bizarre: p-adic numbers. We’ll learn a new way of looking at number, and about a lens through which all triangles become isosceles.”
Lecture 20 – The Notion of Transfinite Numbers
“Although it seems that we’ve looked at all possible worlds of number, we soon find that these worlds open onto a universe of number—and further still. In this lecture, we’ll learn not only how humans arrived at the notion of infinity but how to compare infinities.”
Lecture 21 – Collections Too Infinite to Count
“Now that we are comfortable thinking about the infinite, we’ll look more closely at various collections of numbers, thereby discovering that infinity comes in at least two sizes.”
Lecture 22 – In and Out – The Road to a Third Infinity
“If infinity comes in two sizes, does it come in three? We’ll use set theory to understand how it might. Then we’ll apply this insight to infinite sets as well, a process that leads us to a third kind of infinity.”
Lecture 23 – Infinity – What We Know and What We Don’t
“If there are several sizes of infinity, are there infinitely many sizes of it? Is there a largest infinity? And is there a size of infinity between the infinity of natural numbers and real numbers? We’ll answer two of these questions and learn why the answer to the other is neither provable nor disprovable mathematically.”
Lecture 24 – The Endless Frontier of Number
“Now that we’ve traversed the universe of number, we can look back and understand how the idea of number has changed and evolved. In this lecture, we’ll get a sense of how mathematicians expand the frontiers of number, and we’ll look at a couple of questions occupying today’s number theorists—the Riemann Hypothesis and prime factorization.”

Course No. 1802
The Search for Exoplanets: What Astronomers Know – Joshua Winn
Lecture 4 – Pioneers of Planet Searching

“Chart the history of exoplanet hunting – from a famous false signal in the 1960s, through ambiguous discoveries in the 1980s, to the big breakthrough in the 1990s, when dozens of exoplanets turned up. Astronomers were stunned to find planets unlike anything in the solar system.”
Special Note: This entire series is outstanding! I will eventually be adding most of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 1816
The Inexplicable Universe: Unsolved Mysteries – Neil deGrasse Tyson
Lecture 4 – Inexplicable Physics

“Among the many topics you’ll learn about in this lecture are the discovery of more elements on the periodic table; muon neutrinos, tao particles, and the three regimes of matter; the secrets of string theory (which offers the hope of unifying all the particles and forces of physics); and even the hypothetical experience of traveling through a black hole.”
Special Note: This entire series is outstanding! I will eventually be adding most of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 1830
Cosmology: The History and Nature of Our Universe – Mark Whittle
Lecture 3 – Overall Cosmic Properties

“The universe is lumpy at the scale of galaxies and galaxy clusters. But at larger scales it seems to be smooth and similar in all directions. This property of homogeneity and isotropy is called the cosmological principle.”
Lecture 4 – The Stuff of the Universe
“The most familiar constituents of the universe are atomic matter and light. Neutrinos make up another component. But by far the bulk of the universe—96%—is dark energy and dark matter. The relative amounts of these constituents have changed as the universe has expanded.”
Lecture 6 – Measuring Distances
“Astronomers use a ‘distance ladder’ of overlapping techniques to determine distances in the universe. Triangulation works for nearby stars. For progressively farther objects, observers use pulsating stars, the rotation of galaxies, and a special class of supernova explosions.”
Lecture 8 – Distances, Appearances, and Horizons
“Defining distances in cosmology is tricky, since an object’s distance continually increases with cosmic expansion. There are three important distances to consider: the emission distance, when the light set out; the current distance, when the light arrives; and the distance the light has traveled.”
Lecture 10 – Cosmic Geometry – Triangles in the Sky
“Einstein’s theory of gravity suggests that space could be positively or negatively curved, so that giant billion-light-year triangles might have angles that don’t add up to 180°. This lecture discusses the success at measuring the curvature of the universe in 1998.”
Lecture 11 – Cosmic Expansion – Keeping Track of Energy
“Has the universe’s rate of expansion always been the same? You answer this question by applying Newton’s law of gravity to an expanding sphere of matter, finding that the expansion was faster in the past and slows down over time.”
Lecture 12 – Cosmic Acceleration – Falling Outward
“We investigate why the three great eras of cosmic history—radiation, matter, and dark energy—have three characteristic kinds of expansion. These are rapid deceleration, modest deceleration, and exponential acceleration. The last is propelled by dark energy, which makes the universe fall outward.”
Lecture 13 – The Cosmic Microwave Background
“By looking sufficiently far away, and hence back in time, we can witness the ‘flash’ from the big bang itself. This arrives from all directions as a feeble glow of microwave radiation called the cosmic microwave background (CMB), discovered by chance in 1964.”
Lecture 22 – The Galaxy Web – A Relic of Primordial Sound
“A simulated intergalactic trip shows you the three-dimensional distribution of galaxies in our region of the universe. On the largest scale, galaxies form a weblike pattern that matches the peaks and troughs of the primordial sound in the early universe.”
Lecture 24 – Understanding Element Abundances
“The theory of atom genesis in the interiors of stars is confirmed by the proportions of each element throughout the cosmos. The relative abundances hardly vary from place to place, so that gold isn’t rare just on earth, it’s rare everywhere.”
Lecture 27 – Physics at Ultrahigh Temperatures
“This lecture begins your investigation of the universe during its first second, which is an immense tract of time in nature. To understand what happened, you need to know how nature behaves at ultrahigh energy and density. Fortunately, the physics is much simpler than you might think.”
Lecture 29 – Back to the GUT – Matter and Forces Emerge
“You venture into the bizarre world of the opening nanosecond. There are two primary themes: the birth of matter and the birth of forces. Near one nanosecond, the universe was filled with a dense broth of the most elementary particles. As temperatures dropped, particles began to form.”
Lecture 30 – Puzzling Problems Remain
“Although the standard big bang theory was amazingly successful, it couldn’t explain several fundamental properties of the universe: Its geometry is Euclidean, it’s smooth on the largest scales, and it was born slightly lumpy on smaller scales. The theory of cosmic inflation offers a comprehensive solution.”
Lecture 31 – Inflation Provides the Solution
“This lecture shows how the early universe might enter a brief phase of exponentially accelerating expansion, or inflation, providing a mechanism to launch the standard hot big bang universe. This picture also solves the flatness, horizon, and monopole problems that plagued the standard big-bang theory.”
Lecture 33 – Inflation’s Stunning Creativity
“All the matter and energy in stars and galaxies is exactly balanced by all the negative energy stored in the gravitational fields between the galaxies. Inflation is the mechanism that takes nothing and makes a universe—not just our universe, but potentially many.”
Lecture 34 – Fine Tuning and Anthropic Arguments
“Why does the universe have the properties it does and not some different set of laws? One approach is to see the laws as inevitable if life ever evolves to ask such questions. This position is called the anthropic argument, and its validity is hotly debated.”

Course No. 1866
The Remarkable Science of Ancient Astronomy – Bradley E. Schaefer
Lecture 10 – Origins of Western Constellations
“The human propensity for pattern recognition and storytelling has led every culture to invent constellations. Trace the birth of the star groups known in the West, many of which originated in ancient Mesopotamia. At least one constellation is almost certainly more than 14,000 years old and may be humanity’s oldest surviving creative work.”

Course No. 1872
The Life and Death of Stars – Keivan G. Stassun
Lecture 10 – Eclipses of Stars—Truth in the Shadows
“Investigate the remarkable usefulness of eclipses. When our moon passes in front of a star or one star eclipses another, astronomers can gather a treasure trove of data, such as stellar diameters. Eclipses also allow astronomers to identify planets orbiting other stars.”
Lecture 13 – E = mc2—Energy for a Star’s Life
“Probe the physics of nuclear fusion, which is the process that powers stars by turning mass into energy, according to Einstein’s famous equation. Then examine two lines of evidence that show what’s happening inside the sun, proving that nuclear reactions must indeed be taking place.”
Lecture 14 – Stars in Middle Age
“Delve deeper into the lessons of the Hertzsprung-Russell diagram, introduced in Lecture 9. One of its most important features is the main sequence curve, along which most stars are found for most of their lives. Focus on the nuclear reactions occurring inside stars during this stable period.”
Lecture 19 – Stillborn Stars
“Follow the search for brown dwarfs—objects that are larger than planets but too small to ignite stellar fires. Hear about Professor Stassun’s work that identified the mass of these elusive objects, showing the crucial role of magnetism in setting the basic properties of all stars.”
Lecture 20 – The Dark Mystery of the First Stars
“Join the hunt for the first stars in the universe, focusing on the nearby “Methuselah” star. Explore evidence that the earliest stars were giants, even by stellar standards. They may even have included mammoth dark stars composed of mysterious dark matter.”
Lecture 21 – Stars as Magnets
“Because stars spin like dynamos, they generate magnetic fields—a phenomenon that explains many features of stars. See how the slowing rate of rotation of stars like the sun allows astronomers to infer their ages. Also investigate the clock-like magnetic pulses of pulsars, which were originally thought to be signals from extraterrestrials.”
Lecture 22 – Solar Storms—The Perils of Life with a Star
“The sun and stars produce more than just light and heat. Their periodic blasts of charged particles constitute space weather. Examine this phenomenon—from beautiful aurorae to damaging bursts of high-energy particles that disrupt electronics, the climate, and even life.”

Course No. 1878
Radio Astronomy: Observing the Invisible Universe – Felix J. Lockman
Lecture 5 – Radio Telescopes and How They Work
“Radio telescopes are so large because radio waves contain such a small amount of energy. For example, the signal from a standard cell phone measured one kilometer away is five million billion times stronger than the radio signals received from a bright quasar. Learn how each of these fascinating instruments is designed to meet a specific scientific goal—accounting for their wide variation in form and size.”
Lecture 7 – Tour of the Green Bank Observatory
“The Green Bank Observatory is located within the 13,000-acre National Radio Quiet Zone straddling the border of Virginia and West Virginia. Come tour this fascinating facility where astronomers discovered radiation belts around Jupiter, the black hole at the center of our galaxy, and the first known interstellar organic molecule, and began the search for extra-terrestrial life.”
Lecture 8 – Tour of the Green Bank Telescope
“At 17 million pounds, and with more than 2,000 surface panels that can be repositioned in real time, this telescope is one of the largest moveable, land-based objects ever built. The dish could contain two side-by-side football fields, but when its panels are brought into focus, the surface has errors no larger than the thickness of a business card. Welcome to this rare insider’s view.”
Lecture 9 – Hydrogen and the Structure of Galaxies
“Using the laws of physics and electromagnetic radiation, astronomers can ‘weigh’ a galaxy by studying the distribution of its rotating hydrogen. But when they do this, it soon becomes clear something is very wrong: A huge proportion of the galaxy’s mass has simply gone missing. Welcome to the topsy-turvy world of dark matter, which we now believe accounts for a whopping 90 percent of our own Milky Way.”
Lecture 10 – Pulsars: Clocks in Space
“In the mid-1960s, astronomers discovered signals with predictable periodicity but no known source. In case these signals indicated extraterrestrial life, they were initially labeled LGM, Little Green Men. But research revealed the source of the pulsing radiation to be neutron stars. Learn how a star with a diameter of only a few kilometers and a mass similar to that of our Sun can spin around hundreds of times per second.”
Lecture 11 – Pulsars and Gravity
“A pulsar’s spin begins with its birth in a supernova and can be altered by transfer of mass from a companion star. Learn how pulsars, these precise interstellar clocks, are used to confirm Einstein’s prediction of gravitational waves by observations of a double-neutron-star system, and how we pull the pulsar signal out of the noise.”
Lecture 12 – Pulsars and the 300-Foot Telescope
“Humans constantly use radio transmission these days, for everything from military communications to garage-door openers. How can scientists determine which signals come from Earth and which come from space? Learn how the 300-foot telescope, located in two radio quiet zones, was built quickly and cheaply. It ended up studying pulsars and hydrogen in distant galaxies, and made the case for dark matter.”
Lecture 16 – Radio Stars and Early Interferometers
“When radio astronomers discovered a sky full of small radio sources of unknown origin, they built telescopes using multiple antennas to try to understand them. Learn how and why interferometers were developed and how they have helped astronomers study quasars—those massively bright, star-like objects that scientists now know only occur in galaxies whose gas is falling into a supermassive black hole.”
Lecture 18 – Active Galactic Nuclei and the VLA
“The need for a new generation of radio interferometers to untangle extragalactic radio sources led to the development of the Very Large Array (VLA) in New Mexico. With its twenty-seven radio antennas in a Y-shaped configuration, it gives both high sensitivity and high angular resolution. The VLA provided a deeper and clearer look at galaxies than ever before, and the results were astonishing.”
Lecture 19 – A Telescope as Big as the Earth
“Learn how astronomers use very-long-baseline interferometry (VLBI) with telescopes thousands of miles apart to essentially create a radio telescope as big as the Earth. With VLBI, scientists not only look deep into galactic centers, study cosmic radio sources, and weigh black holes, but also more accurately tell time, study plate tectonics, and more—right here on planet Earth.”
Lecture 20 – Galaxies and Their Gas
“In visible light, scientists had described galaxies as ‘island universes’. But since the advent of radio astronomy, we’ve seen galaxies connected by streams of neutral hydrogen, interacting with and ripping the gases from each other. Now astronomers have come to understand that these strong environmental interactions are not a secondary feature—they are key to a galaxy’s basic structure and appearance.”
Lecture 21 – Interstellar Molecular Clouds
“In the late 1960s, interstellar ammonia and water vapor were detected. Soon came formaldehyde, carbon monoxide, and the discovery of giant molecular clouds where we now know stars and planets are formed. With improvements in radio astronomy technology, today’s scientists can watch the process of star formation in other systems. The initial results are stunning.”
Lecture 22 – Star Formation and ALMA
“With an array of 66 radio antennas located in the high Chilean desert above much of the earth’s atmosphere, the Atacama Large Millimeter/submillimeter Array (ALMA) is a radio telescope tuned to the higher frequencies of radio waves. Designed to examine some of the most distant and ancient galaxies ever seen, ALMA has not only revealed new stars in the making, but planetary systems as well.”
Lecture 23 – Interstellar Chemistry and Life
“Interstellar clouds favor formation of carbon-based molecules over any other kind—not at all what statistical models predicted. In fact, interstellar clouds contain a profusion of chemicals similar to those that occur naturally on Earth. If planets are formed in this rich soup of organic molecules, is it possible life does not have to start from scratch on each planet?”
Lecture 24 – The Future of Radio Astronomy
“Learn about the newest radio telescopes and the exhilarating questions they plan to address: Did life begin in space? What is dark matter? And a new question that has just arisen in the past few years: What are fast radio bursts? No matter how powerful these new telescopes are, radio astronomers will continue pushing the limits to tell us more and more about the universe that is our home.”

Course No. 1884
Experiencing Hubble: Understanding the Greatest Images of the Universe – David M. Meyer
Lecture 5 – The Cat’s Eye Nebula – A Stellar Demise
“Turning from star birth to star death, get a preview of the sun’s distant future by examining the Cat’s Eye Nebula. Such planetary nebulae (which have nothing to do with planets) are the exposed debris of dying stars and are among the most beautiful objects in the Hubble gallery.”
Lecture 7 – The Sombrero Galaxy – An Island Universe
“In the 1920s, astronomer Edwin Hubble discovered the true nature of galaxies as ‘island universes’. Some 80 years later, the telescope named in his honor has made thousands of breathtaking pictures of galaxies. Focus on one in particular—an edge-on view of the striking Sombrero galaxy.”
Lecture 8 – Hubble’s View of Galaxies Near and Far
“Hubble’s image of the nearby galaxy NGC 3370 includes many faint galaxies in the background, exemplifying the telescope’s mission to establish an accurate distance scale to galaxies near and far—along with the related expansion rate of the universe. Discover how Hubble’s success has led to the concept of dark energy.”
Lecture 10 – Abell 2218 – A Massive Gravitational Lens
“One of the consequences of Einstein’s general theory of relativity is evident in Hubble’s picture of the galaxy cluster Abell 2218. Investigate the physics of this phenomenon, called gravitational lensing, and discover how Hubble has used it to study extremely distant galaxies as well as dark matter.”

Course No. 3130
Origin of Civilization – Scott MacEachern
Lecture 36 – Great Zimbabwe and Its Successors
“Few archaeological sites have been subjected to the degree of abuse and misrepresentation sustained by Great Zimbabwe in southeastern Africa. Nevertheless, this lecture unpacks the intriguing history of this urban center and the insights it can provide into the development of the elite.”

Course No. 3900
Ancient Civilizations of North America – Edwin Barnhart
Lecture 12 – The Wider Mississippian World
“After the fall of Cahokia, witness how Mississippian civilization flourished across eastern North America with tens of thousands of pyramid-building communities and a population in the millions. Look at the ways they were connected through their commonly held belief in a three-tiered world, as reflected in their artwork. Major sites like Spiro, Moundville, and Etowah all faded out just around 100 years before European contact, obscuring our understanding.”
Lecture 13 – De Soto Versus the Mississippians
“In 1539, Hernando de Soto of Spain landed seven ships with 600 men and hundreds of animals in present-day Florida. Follow his fruitless search for another Inca or Aztec Empire, as he instead encounters hundreds of Mississippian cities through which he led a three-year reign of terror across the land-looting, raping, disfiguring, murdering, and enslaving native peoples by the thousands.”
Lecture 19 – The Chaco Phenomenon
“Chaco Canyon contains the most sophisticated architecture ever built in ancient North America—14 Great Houses, four Great Kivas, hundreds of smaller settlements, an extensive road system, and a massive trade network. But who led these great building projects? And why do we find so little evidence of human habitation in what seems to be a major center of culture? Answer these questions and more.”
Lecture 24 – The Iroquois and Algonquians before Contact
“At the time of European contact, two main groups existed in the northeast—the hunter-gatherer Algonquian and the agrarian Iroquois. Delve into how the Iroquois created the first North American democracy as a solution to their increasing internal conflicts. Today, we know much of the U.S. Constitution is modeled on the Iroquois’ “Great League of Peace” and its 117 articles of confederation, as formally acknowledged by the U.S. in 1988.”

Course No. 4215
An Introduction to Formal Logic – Steven Gimbel
Lecture 8 – Induction in Polls and Science
“Probe two activities that could not exist without induction: polling and scientific reasoning. Neither provides absolute proof in its field of analysis, but if faults such as those in Lecture 7 are avoided, the conclusions can be impressively reliable.”

Course No. 5006
Capitalism vs. Socialism: Comparing Economic System – Edward F. Stuart
Lecture 13 – French Indicative Planning and Jean Monnet

“Discover why France, a latecomer to industrial capitalism, was vital in shaping influential socialist theories, and how centuries of political upheaval can leave distinct impressions on a nation’s economic history. From the French Revolution to World War II and beyond, France is a strong example of the ways economies are shaped by both internal and external forces.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 7210
The Symphony – Robert Greenberg
Lecture 24 – Dmitri Shostakovich and His Tenth Symphony

“Dmitri Shostakovich was used and abused by the Soviet powers during much of his life. Somehow, he survived—even as his Tenth Symphony made dangerously implicit criticisms of the Soviet government.”

Course No. 7250
Beethoven’s Piano Sonatas – Robert Greenberg
Lecture 4 – The Grand Sonata, Part 2

“Continuing our study of Beethoven’s grand sonatas, we examine Sonata no. 3 in C, no. 3, op. 2, and Sonata no. 4 in E flat, op. 7. In both these works, we see Beethoven’s early artistic declaration that he was not interested in slavishly following the Classical tradition.”
Lecture 15 – The Waldstein and the Heroic Style
“Piano Sonata no. 21 in C, op. 53 (Waldstein) is like no other music written by Beethoven or anyone else. We study this remarkable piece—from its unrelenting opening theme to its breathtaking prestissimo (“as fast as possible”) conclusion.”
Lecture 23 – In a World of His Own
“Beethoven’s last three piano sonatas owe much to his epic Missa Solemnis (“Solemn Mass”) which was also composed in the period 1820–1822. We explore the spiritual and compositional links to the Missa Solemnis, particularly as they relate to sonatas no. 30 in E, op. 109, and no. 31 in A flat, op. 110.”
Lecture 24 – Reconciliation
“Beethoven completed his final piano sonata, no. 32 in C Minor, op. 111, in 1822—five years before his death. Opus 111 seems obviously Beethoven’s valedictory statement for the genre; it ties up loose ends, yet it is so stunningly original that it caps, rather than continues, the composer’s run of 32 sonatas for piano.”

Course No. 7261
Understanding the Fundamentals of Music – Robert Greenberg
Lecture 9 – Intervals and Tunings

“Resuming our focus on pitch, we will turn once more to Pythagoras, and his investigation into what is now known as the overtone series. This paves the way for an examination of intervals, the evolution of tuning systems, and an introduction to the major pitch collections.”

Course No. 7270
The Concerto – Robert Greenberg
Lecture 13 – Tchaikovsky
“Excoriated by colleagues and critics alike, Tchaikovsky’s concerti ultimately triumphed to become cornerstones of the repertoire. This lecture explores his Piano Concerto no. 1 in B flat Minor, op. 23; Piano Concerto no. 2 in G Major, op. 44; and Violin Concerto in D Major, op. 35, arguably his single greatest work and one of the greatest concerti of the 19th century.”
Lecture 14 – Brahms and the Symphonic Concerto
“Johannes Brahms’s compositional style is a synthesis of the clear and concise musical forms and genres of the Classical and Baroque eras, and the melodic, harmonic, and expressive palette of the Romantic era in which he lived. This lecture examines in depth his monumental Piano Concerto no. 2 in B flat Major, op. 83.”

Course No. 8122
Albert Einstein: Physicist, Philosopher, Humanitarian – Don Howard
Lecture 1 – The Precocious Young Einstein

“The aim of these lectures is to explore Einstein the whole person and the whole thinker. You begin with an overview of the course. Then you look at important events in Einstein’s life up to the beginning of his university studies in 1896.”
Lecture 7 – Background to General Relativity
“Special relativity is ‘special’ in the sense that it is restricted to observers moving with constant relative velocity. Einstein wanted to extend the theory to include accelerated motion. His great insight was that such a ‘general’ theory would incorporate the phenomenon of gravity.”
Lecture 19 – Einstein and the Bomb – Science Politicized
“In 1939, Einstein signed a letter to President Roosevelt that launched the Manhattan Project to build the first atomic bomb. Scientists had long advised governments, but this effort represented a fundamental shift in the relationship between science and the state.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 8374
Understanding Russia: A Cultural History – Lynne Ann Hartnett
Lecture 10 – Alexander II, Nihilists, and Assassins

“Focus is on the reign of Alexander II, who ruled Russia from 1855 to 1881. Central to this lecture are three questions: Why did this promising reign end so violently? Did Alexander II shape developments in literature and culture? How could Russia’s last great tsar inaugurate a violent confrontation between the state and its people?”
Lecture 14 – The Rise and Fall of the Romanovs
“Here is the real story behind the Romanov dynasty, from its rise to power in 1613 to its bloody end in 1917—a tale filled with adventure, intrigue, romance, and heartbreak. It was this period that saw the Decembrist revolution, the assassination of Tsar Alexander II, and the machinations of the notorious Grigori Rasputin.”
Lecture 17 – Lenin and the Soviet Cultural Invasion
“Professor Hartnett reveals how Lenin and the Communist Party aimed to win the hearts and minds of the Soviet people through a cultural battle fought on every possible front. See how this battle was won through a militarized economy, propaganda radio, the renaming of streets, and the ‘secular sainthood’ of Lenin.”
Lecture 19 – The Tyrant is a Movie Buff: Stalinism
“Stalin and his cadre aspired to transform everyday Russian life (byt) in ways that brought forth such horrors as collectivization and the gulags. But, as you’ll learn, this was also a period where the creative work and cultural influence of writers, composers, and painters were suppressed by the terrifying mandates of Socialist Realism.”
Lecture 20 – The Soviets’ Great Patriotic War
“By the time World War II ended, the Soviets would lose 27 million men, women, and children from a total population of 200 million. In this lecture, we examine Soviet life during the Great Patriotic War and investigate how culture (including poetry and film) was used in service of the war effort.”
Lecture 21 – With Khrushchev, the Cultural Thaw
“Nikita Khrushchev emerged from the power struggles after Stalin’s death with a daring denunciation of the dictator’s cult of terror and personality. As we examine Khrushchev’s liberalization of culture, we’ll also explore its limits, including the continuation of anti-Semitism from the Stalin era, embraced under the guise of ‘anti-cosmopolitanism’.”
Lecture 22 – Soviet Byt: Shared Kitchen, Stove, and Bath
“What was everyday Soviet life like during the Khrushchev and Brezhnev periods? How and where did people live? How did they spend their leisure time? Answers to these and other questions reveal the degree to which politics affected even seemingly apolitical areas of life.”
Lecture 24 – Soviet Chaos and Russian Revenge
“On December 25, 1991, the Soviet Union came to an end. We follow the road that led to this moment under the policies of perestroika (restructuring the centrally-planned economy) and glasnost (removing rigid state censorship). Then, we conclude with a look at the rise of a new popular leader: Vladimir Putin.”

Course No. 8535
America in the Gilded Age and Progressive Era – Edward T. O’Donnell
Lecture 23 – Over There: A World Safe for Democracy

“As the Progressive Era ends, follow the complex events that led the United States into World War I. Learn how an initial federal policy of neutrality changed to one of “preparedness” and then intervention, amid conflicting public sentiments and government pro-war propaganda. Also trace the after-effects of the war on U.S. foreign policy.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 8580
Turning Points in American History – Edward T. O’Donnell
Lecture 10 – 1786 Toward a Constitution – Shays’s Rebellion

“Who was Daniel Shays? What political and economic dilemmas led to this famous farmer’s rebellion of 1786? Most important: How did this event pave the way for a reconsideration of the Articles of Confederation and the creation of the U. S. Constitution? Find out here.”
Lecture 23 – 1868 Equal Protection—The 14th Amendment
“Many legal scholars and historians have argued that the 14th Amendment, which promises equal protection under the laws, is the most important addition to the Constitution after the Bill of Rights. Here, Professor O’Donnell retells the fascinating story of how this amendment was ratified in 1868—and its turbulent history in the 20th and 21st centuries.”
Special Note: This entire series is outstanding! I will eventually be adding many of the episodes of this course as I rewatch them. (I watched this series before I began keeping track of “best” episodes.)

Course No. 30110
England, the 1960s, and the Triumph of the Beatles – Michael Shelden
Lecture 8 – The Englishness of A Hard Day’s Night
“In summer 1964, the cinematic Beatles vehicle A Hard Day’s Night broke almost every rule in Hollywood at the time. Professor Shelden reveals what lies underneath the film’s surface charm and musical numbers: an overall attitude of irreverence and defiance in the face of authority, and a challenge for audiences to think for themselves.”
Lecture 12 – Hello, Goodbye: The End of the 1960s
“In their last years together, all four of the Beatles seemed headed in new directions as they grew up—and apart. Nevertheless, witness how these final years brought a range of sounds, including protest songs, mystic melodies, anthems of friendship, and an iconic double album called simply, The Beatles, but better known as the ‘White Album.'”

Course No. 60000
The Great Questions of Philosophy and Physics – Steven Gimbel
Lecture 3 – Can Physics Explain Reality?
“If the point of physics is to explain reality, then what counts as an explanation? Starting here, Professor Gimbel goes deeper to probe what makes some explanations scientific and whether physics actually explains anything. Along the way, he explores Bertrand Russell’s rejection of the notion of cause, Carl Hempel’s account of explanation, and Nancy Cartwright’s skepticism about scientific truth.”
Lecture 4 – The Reality of Einstein’s Space
“What’s left when you take all the matter and energy out of space? Either something or nothing. Newton believed the former; his rival, Leibniz, believed the latter. Assess arguments for both views, and then see how Einstein was influenced by Leibniz’s relational picture of space to invent his special theory of relativity. Einstein’s further work on relativity led him to a startlingly new conception of space.”
Lecture 5 – The Nature of Einstein’s Time
“Consider the weirdness of time: The laws of physics are time reversable, but we never see time running backwards. Theorists have proposed that the direction of time is connected to the order of the early universe and even that time is an illusion. See how Einstein deepened the mystery with his theory of relativity, which predicts time dilation and the surprising possibility of time travel.”
Lecture 8 – Quantum States: Neither True nor False?
“Enter the quantum world, where traditional philosophical logic breaks down. First, explore the roots of quantum theory and how scientists gradually uncovered its surpassing strangeness. Clear up the meaning of the Heisenberg uncertainty principle, which is a metaphysical claim, not an epistemological one. Finally, delve into John von Neumann’s revolutionary quantum logic, working out an example.”
Lecture 10 – Wanted Dead and Alive: Schrödinger’s Cat
“The most famous paradox of quantum theory is the thought experiment showing that a cat under certain experimental conditions must be both dead and alive. Explore four proposed solutions to this conundrum, known as the measurement problem: the hidden-variable view, the Copenhagen interpretation, the idea that the human mind “collapses” a quantum state, and the many-worlds interpretation.”
Lecture 11 – The Dream of Grand Unification
“After the dust settled from the quantum revolution, physics was left with two fundamental theories: the standard model of particle physics for quantum phenomena and general relativity for gravitational interactions. Follow the quest for a grand unified theory that incorporates both. Armed with Karl Popper’s demarcation criteria, see how unifying ideas such as string theory fall short.”
Lecture 12 – The Physics of God
“The laws of physics have been invoked on both sides of the debate over the existence of God. Professor Gimbel closes the course by tracing the history of this dispute, from Newton’s belief in a Creator to today’s discussion of the “fine-tuning” of nature’s constants and whether God is responsible. Such big questions in physics inevitably bring us back to the roots of physics: philosophy.”

Course No. 80060
Music Theory: The Foundation of Great Music – Sean Atkinson
Lecture 5 – The Circle of Fifths
“Begin by defining the key of a piece of music, which is simply the musical scale that is used the most in the piece. Also discover key signatures in written music, symbols at the beginning of the musical score that indicate the key of the piece. Then grasp how the major keys all relate to each other in an orderly way, when arranged schematically according to the interval of a fifth.”
Lecture 16 – Hypermeter and Larger Musical Structures
“In listening to music, we sometimes hear the meter differently than the way it’s written on the page. Learn how the concept of hypermeter helps explain this, by showing that when measures of music are grouped into phrases, we often hear a pulse for each measure in the phrase, rather than the pulses within the measure. Explore examples of hypermeter, and how we perceive music as listeners.”