In the third episode we will see Europe by the 17th century taking over from the Middle East as the powerhouse of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was on to discover the mathematics to describe objects in motion. This programme explores the work of Rene Descartes, Pierre Fermat, Isaac Newton, Leonard Euler and Carl Friedrich Gauss. Du Sautoy proceeds to describes René Descartes realisation that it was possible to describe curved lines as equations and thus link algebra and geometry. He talks with Henk J. M. Bos about Descartes. He shows how one of Pierre de Fermat’s theorems is now the basis for the codes that protect credit card transactions on the internet. He describes Isaac Newton’s development of math and physics crucial to understanding the behaviour of moving objects in engineering. He covers the Leibniz and Newton calculus controversy and the Bernoulli family. He further covers Leonhard Euler, the father of topology, and Gauss' invention of a new way of handling equations, modular arithmetic. The further contribution of Gauss to our understanding of how prime numbers are distributed is covered thus providing the platform for Bernhard Riemann's theories on prime numbers. In addition Riemann worked on the properties of objects, which he saw as manifolds that could exist in multi-dimensional space.

For centuries, the precise workings of gravity have confounded the greatest scientific minds - from Newton to Faraday and Einstein - and the idea of controlling gravity has been seen as little more than a fanciful dream. Yet in the mid 1990s, UK defence manufacturer BAE Systems began a ground-breaking project code-named Greenglow. Nasa was simultaneously running its own Breakthrough Propulsion Physics Project". It was concerned with potential space applications of new physics, including concepts like 'faster-than-light travel' and 'warp drives'. Looking into the past and projecting into the future, Horizon explores science's long-standing obsession with the idea of gravity control. It looks at recent breakthroughs in the search for loopholes in conventional physics and examines how the groundwork carried out by Project Greenglow has helped change our understanding of the universe. Gravity control may sound like science fiction, but the research that began with Project Greenglow is very much ongoing, and the dream of flying cars and journeys to the stars no longer seems quite so distant.

Tyson begins the episode by explaining the nature of the speed of light and how much of what is seen of the observable universe is from light emanated from billions of years in the past. Tyson further explains how modern astronomy has used such analyzes via deep time to identify the Big Bang event and the age of the universe. Tyson proceeds to describe how the work of Isaac Newton, William Herschel, and James Clerk Maxwell contributed to understanding the nature of electromagnetic waves and gravitational force, and how this work led towards Albert Einstein's Theory of Relativity, that the speed of light is a fundamental constant of the universe and gravity can be seen as distortion of the fabric of space-time. Tyson describes the concept of dark stars as postulated by John Michell which are not visible but detectable by tracking other stars trapped within their gravity wells, an idea Herschel used to discover binary stars. Tyson then describes the nature of black holes, their enormous gravitational forces that can even capture light, and their discovery via X-ray sources such as Cygnus X-1. Tyson uses the Ship of Imagination to provide a postulate of the warping of spacetime and time dilation as one enters the event horizon of the black hole, and the possibility that these may lead to other points within our universe or others, or even time travel. Tyson ends on noting that Herschel's son, John would be inspired by his father to continue to document the known stars as well as contributions towards photography that play on the same nature of deep time used by astronomers.

The episode begins with Tyson describing how pattern recognition manifested in early civilization as using astronomy and astrology to predict the passing of the seasons, including how the passage of a comet was often taken as an omen. Tyson continues to explain that the origin of comets only became known in the 20th century due to the work of Jan Oort and his hypothesis of the Oort cloud. Tyson then continues to relate the collaboration between Edmond Halley and Isaac Newton in the last part of the 17th century in Cambridge. The collaboration would result in the publication of Philosophiæ Naturalis Principia Mathematica, the first major work to describe the laws of physics in mathematical terms, despite objections and claims of plagiarism from Robert Hooke and financial difficulties of the Royal Society of London. Tyson explains how this work challenged the prevailing notion that God had planned out the heavens, but would end up influencing many factors of modern life, including space flight. Tyson further describes Halley's contributions including determining Earth's distance to the sun, the motion of stars and predicting the orbit of then-unnamed Halley's Comet using Newton's laws. Tyson contrasts these scientific approaches to understanding the galaxy compared to what earlier civilizations had done, and considers this advancement as mankind's first steps into exploring the universe. The episode ends with an animation of the Milky Way and Andromeda galaxies' merging based on the principles of Newton's laws.

This episode explores the wave theory of light as studied by mankind, noting that light has played an important role in scientific progress, with such early experiments from over 2000 years ago involving the camera obscura by the Chinese philosopher Mozi. Tyson describes the work of the 11th century Arabic scientist Ibn al-Haytham, considered to be one of the first to postulate on the nature of light and optics leading to the concept of the telescope, as well as one of the first researchers to use the scientific method. Tyson proceeds to discuss the nature of light as discovered by mankind. Work by Isaac Newton using diffraction through prisms demonstrated that light was composed of the visible spectrum, while findings of William Herschel in the 19th century showed that light also consisted of infrared rays. Joseph von Fraunhofer would later come to discover that by magnifying the spectrum of visible light, gaps in the spectrum would be observed. These Fraunhofer lines would later be determined to be caused by the absorption of light by electrons in moving between atomic orbitals when it passed through atoms, with each atom having a characteristic signature due to the quantum nature of these orbitals. This since has led to the core of astronomical spectroscopy, allowing astronomers to make observations about the composition of stars, planets, and other stellar features through the spectral lines, as well as observing the motion and expansion of the universe, and the existence of dark matter.