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.
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.
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.
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.
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.
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.
At least as early as the 3rd century B.C., Chinese civil servants had to solve problems in surveying and collecting taxes.
Algebra was perfected here in the 9th century by the great mathematician Abu Jafar al-Kwarizmi, whose name would become the word "algorithm."
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.
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.
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.
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.
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.
Independently of Newton, Gottfried Wilhelm Leibniz discovered the techniques of calculus in the 1670s, developing the notational system still used today.
Leonard Euler dominated 18th-century mathematics so thoroughly that his contemporaries believed he had solved all important problems.
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.
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.
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.
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.
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.
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.
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.
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.
This last lecture introduces some of the most promising and important questions in the field and examines mathematical challenges from other disciplines, especially genetics.